pore-scale modelling of wettability alteration in
TRANSCRIPT
Pore-scale Modelling of Wettability Alteration in
Microporous Carbonates
Wissem Kallel
Submitted for the degree of Doctor of Philosophy
Heriot-Watt University
School of Energy, Geoscience, Infrastructure and Society
November 2016
The copyright in this thesis is owned by the author. Any quotation from
the thesis or use of any of the information contained in it must
acknowledge this thesis as the source of the quotation or information.
Abstract
While carbonate reservoirs are recognized to be weakly- to moderately oil-wet at the
core-scale, wettability distributions at the pore-scale remain poorly understood. In
particular, the wetting state of micropores (pores <5 Β΅m in radius) is crucial for assessing
multi-phase flow processes, as microporosity can determine overall pore-space
connectivity. Nonetheless, micropores are usually assumed to be water-wet and their
role in multi-phase flow has often been neglected. However, oil-wet conditions in
micropores are plausible, since oil has been detected within micropores in carbonate
rocks. Modelling the wettability of carbonates using pore network models is
challenging, because of our inability to attribute appropriate chemical characteristics to
the pore surfaces in the presence of the oil phase and over-simplification of the pore
shapes.
First, we carry out an investigation of the prevalent wettability alteration scenario due
to heavy polar compounds (e.g. asphaltenes) adsorption from the oil phase onto the
surface, which occurs strictly after oil invasion. We develop a physically-plausible
wettability distribution that we incorporate in a quasi-static two-phase flow network
model which involves a diversity of pore shapes. The model qualitatively reproduces
patterns of wettability alteration recently observed in microporous carbonates via high-
resolution imaging. To assess the combined importance of pore-space structure and
wettability on petrophysical properties, we consider a homogeneous Berea sandstone
network and a heterogeneous microporous carbonate network, whose disconnected
coarse-scale pores are connected through a sub-network of fine-scale pores. Results
demonstrate that wettability effects are significantly more profound in the carbonate
network, as the wettability state of the micropores controls the oil recovery.
Second, we develop a novel mechanistic wettability alteration scenario that evolves
during primary drainage, involving small polar non-hydrocarbon compounds present in
the oil (e.g. alkylphenols, carbazoles, etc.). We implement a diffusion and adsorption
model for these compounds that triggers a mild wettability alteration from initially
water-wet to more intermediate-wet conditions. This mechanism is incorporated in the
quasi-static pore-network model to which we add a notional time-dependency of the
invasion percolation mechanism. The model qualitatively reproduces experimental
observations where an early rapid wettability alteration occurred during primary
drainage. Additionally, we are able to predict clear differences in the primary drainage
patterns by varying both the strength of wettability alteration and the balance between
the processes of oil invasion and wetting change, which control the initial water
saturation for waterflooding. In fact, under certain conditions, the model results in
higher oil saturations at predefined capillary pressures compared to the conventional
primary drainage. In particular, it leads to the invasion of micropores even at moderate
capillary pressures in the microporous carbonate network. Additionally, the model
results in significant changes in the residual oil saturations after
waterflooding, especially when the wetting state is altered from intermediate-wet to
more oil-wet conditions during ageing.
Acknowledgements
To my supervisors, Rink and Ken, for your continuous support.
To my friends, for the memorable times spent together.
To my parents and brother, for your unconditional love and caring.
To my darling, Yasmine, for your unlimited love and for being the rock
upon which I stand.
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Table of Contents
Chapter 1 : Introduction .................................................................................................... 1
Chapter 2 : Literature review ............................................................................................ 4
2.1 Wettability alteration .............................................................................................. 4
2.1.1 Pore-scale wettability alteration mechanisms ................................................. 5
2.1.2 Factors affecting wettability ............................................................................. 9
2.1.3 Assessing wettability ....................................................................................... 11
2.1.4 Wettability distributions ................................................................................. 16
2.1.5 Effect of wettability on the residual oil saturation ......................................... 19
2.2 Carbonates and microporosity .............................................................................. 25
2.2.1 Composition and structure ............................................................................. 26
2.2.2 Wettability ...................................................................................................... 28
2.2.3 Effect of wettability on oil recovery ............................................................... 31
2.2.4 Multi-scale modelling ..................................................................................... 33
2.3 Discussion .............................................................................................................. 34
Chapter 3 : Pore network model ..................................................................................... 36
3.1 Introduction ........................................................................................................... 36
3.2 Pore Network modelling tool ................................................................................ 37
3.2.1 Pore shapes ..................................................................................................... 37
3.2.2 Fluid configuration .......................................................................................... 40
3.2.3 Pore-level displacements ................................................................................ 41
3.2.4 Flooding cycle ................................................................................................. 45
3.2.5 Network and phases connectivity .................................................................. 50
3.3 Input networks ...................................................................................................... 51
3.3.1 Berea sandstone network ............................................................................... 52
3.3.2 Multiscale carbonate network ........................................................................ 52
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3.4 Conclusion ............................................................................................................. 54
Chapter 4 : Scenario 1 β Wettability alteration following ageing ................................... 56
4.1 Introduction ........................................................................................................... 56
4.2 Model description ................................................................................................. 56
4.3 Results and discussion ........................................................................................... 58
4.3.1 Berea sandstone network ............................................................................... 59
4.3.2 Carbonate network ......................................................................................... 62
4.4 Discussion .............................................................................................................. 70
4.5 Conclusions ............................................................................................................ 71
Chapter 5 : Scenario 2 β Wettability alteration starting during primary drainage ......... 73
5.1 Introduction ........................................................................................................... 73
5.2 Model Description ................................................................................................. 74
5.2.1 Transport through oil invasion ....................................................................... 76
5.2.2 Transport through diffusion ........................................................................... 78
5.2.3 Wettability alteration ..................................................................................... 82
5.3 Results and discussion ........................................................................................... 87
5.3.1 Berea sandstone network ............................................................................... 88
5.3.2 Carbonate network ....................................................................................... 111
5.4 Conclusions .......................................................................................................... 123
Chapter 6 : Conclusions and future work ..................................................................... 129
6.1 Summary and main conclusions .......................................................................... 129
6.1.1 Scenario 1- Wettability alteration following ageing ..................................... 129
6.1.2 Scenario 2- Wettability alteration starting during primary drainage ........... 130
6.2 Discussion ............................................................................................................ 131
6.3 Future work ......................................................................................................... 133
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List of Tables
Table 3.1: Main properties of the Berea network. ......................................................... 52
Table 3.2: Main properties of the carbonate network. .................................................. 53
Table 4.1: Base case parameters for Scenario 1 simulations.......................................... 58
Table 5.1: Base case parameters for Scenario 2 simulations in the Berea network. Only
the last three parameters are varied during the sensitivity study. ................................ 88
Table 5.2: Base case parameters for Scenario 2 simulations in the carbonate network.
Only the last three parameters are varied during the sensitivity study. ...................... 112
Table 6.1: Typical computational times for the two developed scenarios run on the Berea
and carbonate networks using the base case parameters (shown in Table 4.1, Table 5.1
and Table 5.2). Note that the computer has an Intel core i7 processor, and that multiple
simulations can be run simultaneously without altering the computational efficiency
....................................................................................................................................... 132
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List of Figures
Figure 2.1: Wettability illustrated using a pure water drop at equilibrium on a smooth
solid surface, surrounded by pure oil. If ΞΈ < 90Β°, the surface is water-wet and the oil
drop spreads on the surface. Otherwise, if ΞΈ > 90Β°, the surface is oil-wet and the oil
drop forms a bead, minimizing the contact with the solid. .............................................. 4
Figure 2.2: Evolution of thin film stability during oil invasion, illustrated on an example
of disjoining pressure isotherm Πd, after Hirasaki (1991). ............................................... 6
Figure 2.3: (a) Plot of the normalized concentrations of polar compounds in the
produced oil (mobile) function of the elution time; (b) Plot of the concentration of βp-
cresolβ in the core extract oil (immobile) relative to the distance along the core; ESEM
images showing water spreading trends from (c) the outlet and (d) inlet of the core
(Bennett et al., 2004). ....................................................................................................... 8
Figure 2.4: Structural position effect on fluid saturations and wettability within a typical
reservoir. In the water and oil zones, all pores are water-filled & water-wet and oil-filled
& oil-wet, respectively. In the transition zone, the smallest and largest pores are water-
filled & water-wet and oil-filled & oil-wet, respectively. However, some intermediate-
sized pores may be oil-filled & water-wet, as the thin water film may resist the moderate
capillary pressure exerted. .............................................................................................. 11
Figure 2.5: An example of a measured contact angle (53Β°); it is the complement of the
traced angle (pink arc, 127Β°) measured through the CO2 (black, less dense phase)
(Andrew et al., 2014)....................................................................................................... 13
Figure 2.6: Calculation of the Amott indices to water,Iw, and oil, Io, and their
combination through the Amott-Harvey index IAH, as well as the USBM index, IUSBM.
These depend on the water saturations, Sj, at the end of the flood j ( j = 1 β primary
drainage, 2 - spontaneous imbibition, 3 β forced imbibition, 4 β spontaneous drainage,
5 - forced drainage) (Dixit et al., 2000). .......................................................................... 14
Figure 2.7: 10Β΅m x10Β΅m AFM image (scanned under water) of a calcite surface aged in
crude oil for (a) 2 days and (b) 21 days. Note the presence of a continuous adsorbate
layer covering the surface in (a), which gets thicker and more stable by increasing the
ageing time in (b) with the presence of large particles with irregular morphologies
(asphaltenes aggregates) (Morrow and Buckley, 2006). ................................................ 15
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Figure 2.8: The observed oil-water-rock contacts which fall in three different categories:
(a) a conventional three-phase contact line; (b) a thin water film keeping the solid
surface from direct contact with oil; and (c) a local oil-rock pinning at geometrical and
chemical heterogeneities (asperities) (Schmatz et al., 2015). ....................................... 16
Figure 2.9: Wettability classification system, after Ryazanov (2012). ............................ 17
Figure 2.10: Relationship between IAH and IUSBM indices that were measured from the
same core sample by different authors. The MWL, FW and MWS lines are derived
analytically (Dixit et al., 2000). ........................................................................................ 18
Figure 2.11: Residual oil saturation as a function of the Amott-Harvey index for three
different crude oils and different PV of water injected (Jadhunandan and Morrow, 1995).
......................................................................................................................................... 20
Figure 2.12: Residual oil saturation as a function of the Amott-Harvey index from 30
sandstone reservoirs (Skauge and Ottesen, 2002). ........................................................ 21
Figure 2.13: Ultimate residual oil saturation as a function of contact angle (left) and oil-
wet fraction (right) (Blunt, 1998). Note that βSnap-offβ, βNucleated wettingβ, βNon-
wetting advanceβ and βOil layersβ correspond to the regimes i., ii., iii. and iv. as described
in the text, respectively. .................................................................................................. 23
Figure 2.14: Comparison between PNM simulation and experimental data from
Jadhunandan and Morrow (1995) of the oil recovery (the Fraction of Oil In Place) as a
function of wettability (Amott-Harvey index) following a 3PV injection of water (Zhao et
al., 2010). ......................................................................................................................... 24
Figure 2.15: Comparison between PNM simulation and experimental data from
Jadhunandan and Morrow (1995) of the recovery factor (RF) as a function of wettability
(Amott-Harvey index) at breakthrough (BT), and following 3PV and 20PV injection of
water. The infinite (inf) PV case was also included on each figure (Ryazanov et al., 2014).
......................................................................................................................................... 25
Figure 2.16: SEM images from microporous carbonates (a) displaying the rhomboidal
micrite crystals and (b) showing the structure of the microporous network and its
connection to the larger mesopore (using epoxy resin cast) (Harland et al., 2015). ..... 27
Figure 2.17: Illustration of the four different microporosity types observed by Cantrell
and Hagerty (1999): (a) microporous grains, where the fully micritised grains seem like
sponges; (b) microporous matrix, consisting of a network of connected micropores; (c)
microporous fibrous to bladed cements, where micropores are found between cement
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blades; and (d) microporous equant cements, with the appearance of micropores
between cement crystals. ............................................................................................... 27
Figure 2.18: FESEM imaging (250 nm scale bars) of the oil deposits (i.e. the footprint of
the wettability alteration) on calcite micro-particles in carbonate rocks (Marathe et al.,
2012). .............................................................................................................................. 30
Figure 2.19: Oil recovery for three different carbonate rocks (two βgrainstonesβ in black
and blue, and a βboundstoneβ in red) as a function of the Amott-Harvey index. Three
wettability states (MXW, MXW-F and MXW-DF, refer to text) are compared to the
VSWW base case. WW: water-wet; IW: intermediate-wet (WWW: weakly water-wet;
NW: neutral-wet; WOW: weakly oil-wet); OW: oil-wet (Tie and Morrow, 2005). ......... 32
Figure 3.1: Different workflows leading from a core sample to relative permeability and
capillary pressure curves, after Ryazanov (2012). .......................................................... 36
Figure 3.2: An example of a n-cornered star shape (n = 5), with inscribed and hydraulic
radii Rins and Rh, respectively, and half-angle Ξ³. .......................................................... 39
Figure 3.3: Shape factor, G, as a function of dimensionless hydraulic radius, H, for some
cross-sectional shapes. The right and left boundaries represent the shapes theoretical
limiting (G, H) pairs (Ryazanov, 2012). ........................................................................... 40
Figure 3.4: Possible cross-sectional fluid configurations following primary drainage and
imbibition for regular n-cornered star shapes (represented by equilateral triangles for
simplicity). Oil is red; water is blue and surfaces of altered wettability are brown
(Ryazanov, 2012). ............................................................................................................ 41
Figure 3.5: Pore-level displacement mechanisms, (a) piston-like, (b) snap-off, (c) and (d)
I1 and I2 PBF events, with 1 and 2 adjacent pores filled with the non-wetting oil phase,
respectively, as observed from micromodel experiments by Lenormand et al. (1983). 42
Figure 3.6: Illustration of different water invasion patterns, depending whether water in
the corners is (a) present or (b) absent. In the first case, βbypassingβ is likely to occur,
which tends to create trapped oil clusters, as opposed to an efficient oil sweep from the
inlet for the second case. Note that water (blue) displaces oil (red) from the inlet (left)
to the outlet (right) of a 2D regular network. ................................................................. 43
Figure 3.7: Illustration of the oil-layer configuration at the corner of a triangular cross-
section, bounded by inner and outer arc menisci AM1 and AM2, with contact angles
ΞΈhand ΞΈa, respectively. Lsdris the distance of AM1 from the corner edge. Ξ³ is the half-
angle. Oil is red, water is blue and surfaces of altered wettability are brown. .............. 45
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Figure 3.8: An example of Euler number calculations carried out on an illustrative 2D
regular cubic network consisting of spherical nodes and tubular bonds. Periodic
boundary conditions apply between the bottom and top. Oil (red) displaced water (blue)
from the inlet (left) to the outlet (right); the βnotionalβ boundary nodes are illustrated
as dashed hollow circles. The network is a single object (Ξ²0 =1) with 7 redundant loops
(Ξ²1 = 7). Further, NN = 15 (9 actual + 6 boundary nodes) and NB = 21; hence ΟNet =
Ξ²0-Ξ²1 = NN-NB = -6. Similarly, for the oil phase, (Ξ²0oil, Ξ²1oil) = (3, 2) and
(NNoil, NBoil) = (14, 13), hence Οoil = 1. ................................................................... 51
Figure 3.9: Berea network: (a) 3D representation and (b) pore size (inscribed radius) and
shape distributions. ......................................................................................................... 52
Figure 3.10: (a) Statistically generated fine network extracted from a micro-CT image at
2.86 ΞΌm resolution, (b) coarse network extracted from another micro-CT image at
14.29 ΞΌm resolution and (c) the resulting integrated two-scale network. Note that both
images are derived from the same microporous carbonate dataset. ............................ 53
Figure 3.11: Carbonate network pore size (inscribed radius) and shape distributions. . 54
Figure 3.12: Comparison between the connectivity functions of the carbonate and Berea
networks, where ΟNet/VNet is the Euler number of the network divided by its total
volume. ............................................................................................................................ 54
Figure 4.1: Equivalent pore wall curvature assignment for (a) n-cornered Polygon and (b)
n-cornered Star shape, where Rins and Rins' denote the original and new inscribed
radii, respectively; rc denotes the radius of curvature and Ο the angle between the
tangent to the newly obtained (red) curved shape at a vertex and the line connecting
the vertex to the centre (Ο coincides with the corner half-angle, Ξ³, for the original
shapes). ........................................................................................................................... 57
Figure 4.2: Primary drainage Pc curve for the Berea network. ..................................... 59
Figure 4.3: Pore occupancies for the Berea network shown on the pore size distribution
following PD to different Swi values. ............................................................................. 59
Figure 4.4: Different wettability distributions shown on the pore size distributions for
the Berea network at fow = 0.5, established after PD for the base case; red: oil-wet,
blue: water-wet. Darker blue (respectively red) indicates stronger water- (respectively
oil) -wetness. ................................................................................................................... 60
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Figure 4.5: Waterflood residual oil saturations as a function of oil-wet fractions for the
Berea network (a) for the different wettability distributions, and (b) for the AW
distribution, with and without oil layers. ........................................................................ 61
Figure 4.6: Pore occupancies at the end of the waterflood for the different wettability
distributions shown on the pore size distribution for the Berea network at fow = 0.5.
......................................................................................................................................... 61
Figure 4.7: Waterflood residual oil saturations for the AW distribution in the Berea
network as a function of (a) Swi and (b) fow. ................................................................ 62
Figure 4.8: Primary drainage Pc curve for the carbonate network. .............................. 63
Figure 4.9: Pore occupancies for the carbonate network shown on the pore size
distribution following PD to different Swi values. .......................................................... 63
Figure 4.10: Different wettability distributions shown on the pore size distributions for
the carbonate network at fow = 0.5, established after PD for the base case; red: oil-
wet, blue: water-wet. ...................................................................................................... 64
Figure 4.11: (a) Pc curves, (b) enlarged Pc curves (red box), (c) Kr curves and (d)
fractional flow of water, Fw, curves after waterflood for the different wettability
distributions at fow = 0.5 for the carbonate network. ................................................. 65
Figure 4.12: Pore occupancies at the end of the waterflood for the different wettability
distributions shown on the pore size distribution for the carbonate network at fow =
0.5. ................................................................................................................................... 66
Figure 4.13: Evolution of the oil phase connectivity (normalised Euler number) during
waterflood for the different wettability distributions at fow = 0.5 in the carbonate
network. .......................................................................................................................... 67
Figure 4.14: Pc curves after waterflood for different oil-wet fractions for the carbonate
network. .......................................................................................................................... 67
Figure 4.15: Waterflood residual oil saturations as a function of oil-wet fractions for the
carbonate network (a) for the different wettability distributions, and (b) for the AW
distribution, with and without oil layers. ........................................................................ 68
Figure 4.16: Waterflood residual oil saturation for the AW distribution in the carbonate
network as a function of (a) Swi and (b) fow. ................................................................ 69
Figure 4.17: Pore occupancies at the end of the waterflood for different Swi values
shown on the pore size distribution for the carbonate network at fow = 0.5. ............. 69
Figure 4.18: Contour chart describing the relationship between Πcrit, Swi and fow. .. 70
ix
Figure 5.1: Illustration of the separation in time of the oil invasion events, where the
cumulative volume of pores invaded linearly increases over time based on the assumed
flow rate, Q; and the incorporation of a diffusion/adsorption model for polar
compounds using discrete time steps, βtTR , during the period of time ti +
1-ti separating two successive pore invasion events. .................................................... 75
Figure 5.2: Illustration of the transport of polar compounds within a pore i with
triangular cross-section during oil invasion (at time ti); as well as their distribution
between the phases within the same pore and their adsorption onto the surface, which
are assumed to happen right after invasion (at time ti +). Note that Cio and Ciware the
mobile concentrations of polar compounds in the oil phase and water phase,
respectively, and Ξi is the corresponding adsorption level of polar compounds per unit
area. ................................................................................................................................ 78
Figure 5.3: Illustration of the diffusion process between (a) two adjacent pores sharing
the same bulk (oil) phase and (b) an oil-filled pore adjacent to a water-filled pore (cross-
phase diffusion) with the partitioning of polar compounds from the oil to the water
phase at the interface and their diffusion within the water phase. Note that Cio and
Ciware the mobile concentrations of polar compounds in the oil phase and water phase,
respectively; Jji and Lji are the diffusion flux and length, respectively, from pore j to pore
i. Note as well the colour key provided in Figure 5.2. .................................................... 81
Figure 5.4: Illustration of the corner water shrinking (and possible collapse) due to the
contact angle, ΞΈ, increase within a pore with triangular cross-section and half-angle Ξ³.
Note the colour key provided in Figure 5.2. ................................................................... 84
Figure 5.5: Evolution of thin film stability during polar compounds adsorption (increasing
Ξ i.e. decreasing Ξ± = cosΞΈ, where Ξ and ΞΈ are the adsorption level of polar compounds
per unit area and contact angle, respectively, linked through Equation (5.14)) at fixed
Pc, illustrated on an example of disjoining pressure isotherm Πd. ............................... 85
Figure 5.6: Network-scale representation of the PD/WE model, involving the time-
dependent oil invasion coupled with the transport model for polar compounds; ti is the
invasion time of pore i, in accordance with the example in Figure 5.1. Oil displaces water
from the inlet (left) to the outlet (right) of a 2D regular network. Note the colour key
provided in Figure 5.2. .................................................................................................... 86
Figure 5.7: Pore occupancies for the Berea network shown on the x-axis (parallel to flow,
from inlet (left) to outlet (right)) following a conventional PD at Pcmax = 6600 Pa (i.e.
x
PD/WE for the base case parameters β with P = 0 or ΞΈmax = 0Β°). The simulation
reached Swi = 0.2 at Pcmax after time tf = 53 min. ................................................... 89
Figure 5.8: Pore occupancies (upper) and (altered) contact angles (lower) for the Berea
network shown on the x-axis (parallel to flow, from inlet (left) to outlet (right)) following
PD/WE for the base case parameters after (a) t1/2 = 31 min (at which Sw β Swi +
12) and (b) tf = 315 min (Swi = 0.22 at Pcmax). ........................................................ 91
Figure 5.9: Evolution of the oil and water phases connectivities (normalised Euler
numbers) during PD/WE for the base case parameters in the Berea network. ............. 92
Figure 5.10: Pore occupancies (upper) and (altered) contact angles (lower) for the Berea
network shown on the x-axis (parallel to flow, from inlet (left) to outlet (right)) following
PD/WE for the base case parameters β with ΞΈmax = 30Β° β after t1/2 = 29 min (at
which Sw β Swi + 12). .................................................................................................. 93
Figure 5.11: Evolution of the oil and water phases connectivities (normalised Euler
numbers) during PD/WE for the base case parameters β with ΞΈmax = 30Β° β in the Berea
network. .......................................................................................................................... 93
Figure 5.12: Swi as a function of PV for the base case parameters β with different ΞΈmax
imposed, in the Berea network. ...................................................................................... 94
Figure 5.13: Pore occupancies for the Berea network shown on the pore-size distribution
following PD/WE for the base case parameters β with (a) ΞΈmax = 0Β° (Swi = 0.2); (b)
ΞΈmax = 30Β° (Swi = 0.14); (c) ΞΈmax = 60Β° (Swi = 0.05) and (d) ΞΈmax = 80Β° (Swi =
0.22). ............................................................................................................................... 95
Figure 5.14: (a) Pore occupancies (upper) and (altered) contact angles (lower) for the
Berea network shown on the x-axis (parallel to flow, from inlet (left) to outlet (right))
following PD/WE for the base case parameters β with Ξmax = 1.5 mgm2 β after t1/2 =
32 min (at which Sw β Swi + 12) and (b) Pore occupancies (only) at tf = 433 min
(Swi = 0.09 at Pcmax). .................................................................................................. 96
Figure 5.15: Evolution of the oil and water phases connectivities (normalised Euler
numbers) during PD/WE for the base case parameters β with Ξmax = 1.5 mgm2 β in
the Berea network. ......................................................................................................... 96
Figure 5.16: Swi as a function of PV for the base case parameters β with different
Ξmax[ mgm2] imposed β and the βNo Adsorptionβ case, in the Berea network. ......... 97
Figure 5.17: The evolution in time of the average mobile concentration of polar
compounds in the oil phase at the outlet bonds, normalised by C0, for the base case
xi
parameters β with different Ξmax[ mgm2] imposed β and the βNo partitioningβ case, in
the Berea network. ......................................................................................................... 98
Figure 5.18: Pore occupancies for the Berea network shown on the pore-size distribution
following PD/WE for the base case parameters β with (a) Ξmax = 0.03 (Swi = 0.42) and
(b) Ξmax = 1.5 mgm2 (Swi = 0.09). ............................................................................ 99
Figure 5.19: The resulting Swi as a function of Ξmax [ mgm2] following the PD/WE
model in the Berea network for the base case parameters β with different ΞΈmax values.
FWB and SWB are the limiting fast wetting and slow wetting boundaries, respectively.
....................................................................................................................................... 100
Figure 5.20: Distribution in the oil column in the Berea network following the PD/WE
model for the base case parameters β with different combinations of ΞΈmax and
Ξmax[ mgm2] values. FWB and SWB are the limiting fast wetting and slow wetting
boundaries, respectively. .............................................................................................. 101
Figure 5.21: The resulting Swi as a function of Q [ m3s] following the PD/WE model in
the Berea network for the base case parameters β with different ΞΈmax values. FWB and
SWB are the limiting fast wetting and slow wetting boundaries, respectively. ........... 102
Figure 5.22: Illustration of the alternative time-dependent oil invasion model, similar to
that described in Figure 5.1 , but here the capillary pressure increases linearly over time
at constant P [Pa. s-1] until it reaches the predefined maximum capillary pressure,
Pcmax. Note that Pentryi denotes the entry pressure of pore i. ................................ 103
Figure 5.23: The resulting Swi as a function of Ξmax [ mgm2] following the PD/WE
model using an alternative time-dependent capillary pressure model in the Berea
network. The base case parameters are used β except for Q replaced by P = 10 Pa. s-1β
with different ΞΈmax values. FWB and SWB are the limiting fast wetting and slow wetting
boundaries, respectively. .............................................................................................. 104
Figure 5.24: Pore occupancies at the end of the waterflood after PD/WE for the base
case parameters β with (a) ΞΈmax = 0Β°(Soi = 0.8; Sor = 0.48), (b) ΞΈmax = 30Β° (Soi =
0.86; Sor = 0.4), (c) ΞΈmax = 60Β° (Soi = 0.95; Sor = 0.4) and (b) ΞΈmax = 80Β° (Soi =
0.78; Sor = 0.44) , shown on the pore size distribution for the Berea network (no
ageing). Note the colour key (top) for the various pore-level displacements. ............. 105
Figure 5.25: Waterflood (a) residual oil saturations and (b) water phase connectivity
(normalised Euler number) as a function of Soi, in the Berea network, following the
xii
application of the PD/WE model for the base case parameters β with varying height h
for the different ΞΈmax values β and no subsequent ageing (fow = 0). ...................... 107
Figure 5.26: Wettability distribution (AW at fow = 0.5) after PD/WE for the base case
parameters β with (a) ΞΈmax = 0Β°, (b) ΞΈmax = 30Β°, (c) ΞΈmax = 60Β° and (b) ΞΈmax =
80Β°, shown on the pore size distribution for the Berea network. ............................... 108
Figure 5.27: Pore occupancies at the end of the waterflood after PD/WE for the base
case parameters β with (a) ΞΈmax = 0Β°(Soi = 0.8; Sor = 0.54), (b) ΞΈmax = 30Β° (Soi =
0.86; Sor = 0.55), (c) ΞΈmax = 60Β° (Soi = 0.95; Sor = 0.44) and (b) ΞΈmax =
80Β° (Soi = 0.78; Sor = 0.36), shown on the pore size distribution for the Berea
network (AW distribution at fow = 0.5). Note the colour key (top) for the various pore-
level displacements. ...................................................................................................... 109
Figure 5.28: Waterflood residual oil saturations in the Berea network as a function of Soi
following the PD/WE model for the base case parameters β with varying height h for the
different ΞΈmax values β and subsequent ageing (AW distribution at fow = 0.5). ...... 109
Figure 5.29: Pore occupancies for the carbonate network shown on the pore-size
distribution following PD/WE for the base case parameters β with (a) ΞΈmax = 0Β° (Swi =
0.5); (b) ΞΈmax = 30Β° (Swi = 0.46); (c) ΞΈmax = 60Β° (Swi = 0.32) and (d) ΞΈmax = 80Β°
(Swi = 0.22). ................................................................................................................ 113
Figure 5.30: (a) Pore occupancies and (b) wettability alteration for the carbonate
network shown on the pore-size distribution following PD/WE for the base case
parameters stopped at a predefined Sw = 0.5. .......................................................... 113
Figure 5.31: Pore occupancies for the carbonate network shown on the pore-size
distribution following PD/WE for the base case parameters β with (a) Ξmax = 0.1
(Swi = 0.26) and (b) Ξmax = 1.4 mgm2 (Swi = 0.13). ............................................. 114
Figure 5.32: (a) Pore occupancies and (b) wettability alteration for the carbonate
network shown on the pore-size distribution following PD/WE at Ξmax =
1.4 mgm2 and ΞΈmax = 80Β°, stopped at a predefined Sw = 0.5. ............................... 114
Figure 5.33: The resulting Swi as a function of Ξmax [ mgm2] following the PD/WE
model in the carbonate network for the base case parameters β with different ΞΈmax
values. FWB and SWB are the limiting fast wetting and slow wetting boundaries,
respectively. .................................................................................................................. 115
Figure 5.34: Comparison between the Berea and carbonate networksβ evolution of the
water phase connectivity (normalised Euler number) during PD/WE at ΞΈmax = 80Β° at
xiii
the highest point in the oil column for the fast wetting boundary (FWB) limiting case.
....................................................................................................................................... 115
Figure 5.35: The resulting (volumetric) fraction of micropores invaded by oil as a function
of Ξmax [ mgm2] following the PD/WE model in the carbonate network for the base
case parameters β with different ΞΈmax values. FWB and SWB are the limiting fast
wetting and slow wetting boundaries, respectively. .................................................... 116
Figure 5.36: Distribution in the oil column in the carbonate network following the PD/WE
model for the base case parameters β with different combinations of ΞΈmax and
Ξmax[ mgm2] values. FWB and SWB are the limiting fast wetting and slow wetting
boundaries, respectively. .............................................................................................. 117
Figure 5.37: Pore occupancies at the end of the waterflood after PD/WE for the base
case parameters β at h = 27m and (a) ΞΈmax = 0Β°(Soi = 0.98; Sor = 0.76), (b) ΞΈmax =
30Β° (Soi = 0.98; Sor = 0.74), (c) ΞΈmax = 60Β° (Soi = 0.95; Sor = 0.7) and (b) ΞΈmax =
80Β° (Soi = 0.79; Sor = 0.63) , shown on the pore size distribution for the carbonate
network (no ageing). Note the colour key (top) for the various pore-level displacements.
....................................................................................................................................... 118
Figure 5.38: Waterflood (a) residual oil saturations and (b) water phase connectivity
(normalised Euler number) as a function of Soi, in the carbonate network, following the
application of the PD/WE model for the base case parameters β with varying height h
for the different ΞΈmax values β and no subsequent ageing (fow = 0). ...................... 119
Figure 5.39: Wettability distribution (AW at fow = 0.5) after PD/WE for the base case
parameters β at h = 27m and (a) ΞΈmax = 0Β°, (b) ΞΈmax = 30Β°, (c) ΞΈmax = 60Β° and (b)
ΞΈmax = 80Β°, shown on the pore size distribution for the carbonate network. ......... 120
Figure 5.40: Pore occupancies at the end of the waterflood after PD/WE for the base
case parameters β at h = 27m and (a) ΞΈmax = 0Β°(Soi = 0.98; Sor = 0.3), (b) ΞΈmax =
30Β° (Soi = 0.98; Sor = 0.26), (c) ΞΈmax = 60Β° (Soi = 0.95; Sor = 0.17) and (b)
ΞΈmax = 80Β° (Soi = 0.79; Sor = 0.17) , shown on the pore size distribution for the
carbonate network (AW distribution at fow = 0.5). Note the colour key (top) for the
various pore-level displacements. ................................................................................ 121
Figure 5.41: Waterflood residual oil saturations in the carbonate network as a function
of Soi following the PD/WE model for the base case parameters β with varying height h
for the different ΞΈmax values β and subsequent ageing (AW distribution at fow = 0.5).
....................................................................................................................................... 121
xiv
Figure 5.42: Contour chart describing the relationship between ΞΈmax, h and fow at a
chosen Πcrit = 33kPa (for which fow = 0.5 at ΞΈmax = 40Β°). .................................. 122
Figure 5.43: Flowchart describing the complex interaction between the input
parameters (in orange), the modelled processes (PD/WE, Ageing and Waterflood, in
blue) and their output results (in green). ..................................................................... 127
Figure 5.44: Representation of the PD/WE model and its effects on oil invasion by a snap-
shot of a simplified 2D 4x4 carbonate network of pores with square cross-sections (half-
angle Ξ³) and two distinct pore sizes: small micropores joining up disconnected larger
macropores. Note that periodic boundary conditions apply between the bottom and
top. The network is initially water-filled and perfectly water-wet (initial contact angle
ΞΈ = 0Β°), then oil displaces water from the inlet (left) to the outlet (right). For simplicity,
we assume that only the macropores can be invaded at the initial wetting conditions
and at the chosen (low) predefined maximum capillary pressure, Pcmax. Note that
PentryΞΈ of a water-filled pore corresponds to its entry pressure at ΞΈ. ....................... 128
xv
List of Publications
Journal papers:
Kallel W., Van Dijke, M. I. J., Sorbie, K. S., Wood, R., Jiang, Z., & Harland, S. (2015).
Modelling the effect of wettability distributions on oil recovery from microporous
carbonate reservoirs. Advances in Water Resources, 1-12.
10.1016/j.advwatres.2015.05.025.
Harland, S. R., Wood, R. A., Curtis, A., Van Dijke, M. I. J., Stratford, K., Jiang, Z., Kallel W.,
& Sorbie, K. S. (2015). Quantifying flow in variably wet microporous carbonates using
object-based geological modelling and both lattice-Boltzmann and pore network fluid
flow simulations. AAPG Bulletin, 99(10), 1827β1860. 10.1306/04231514122.
Kallel W., Van Dijke, M. I. J., Sorbie & K. S., Wood, R (2016). Pore-scale Modelling of
Wettability Alteration during Primary Drainage (submitted to Water Resources
Research).
Conference presentations:
Modelling Wettability in Microporous Carbonates, International Conference on
Computational Methods in Water Resources (CMWR), Stuttgart, June 2014.
Pore-scale Modelling of Wettability Alteration during Primary Drainage, International
Conference on Porous Media (InterPore), Padua, Italy, May 2015.
Pore-scale Wettability Evolution Model during Primary Drainage, International
Conference on Porous Media (InterPore), Cincinnati, OHIO, USA, May 2016.
xvi
Nomenclature
COBR Crude oil/brine/rock system;
CTS Circle-Triangle-Square cross-sectional shape characterisation
of pores;
π The curvature of the cross-sectional shape of a pore;
πͺπ(π), πͺπ°(π) and
πͺπ
The mobile concentrations of polar compounds in a given pore
at time t in the oil and water phases, respectively (πΆπ(t) and
πΆw(t)) and the inlet concentration of polar compounds (πΆ0);
D Fickβs diffusion coefficient;
(FE)SEM (Field-Emission) Scanning Electron Microscopy;
FWB, SWB Fast-wetting boundary (FWB) and slow-wetting boundary
(SWB), which are limiting cases for the PD/WE model;
πππ The volumetric fraction of oil-wet pores among all the pores in
the network (πππ€ β [0,1]);
π Height in the oil column, linked to a fixed capillary pressure;
IP Invasion percolation mechanism;
π°π¨π―, π°πΌπΊπ©π΄ The Amott-Harvey index (πΌπ΄π») and the United States Bureau of
Mines index (πΌπππ΅π);
K The Langmuirβs adsorption constant;
π²ππ, π²ππ The oil and water relative permeabilities, respectively;
micro-CT micro X-ray computed tomography;
MWL, MWS, FW
and AW
Inter-pore wettability distributions: the common Mixed-Wet
Large (MWL), Mixed-Wet Small (MWS) and Fractionally-Wet
(FW); and the developed physically-based Altered-Wet (AW,
Chapter 4);
OWC The oil-water contact i.e. the highest level in the oil column at
which the oil saturation is zero;
P The partitioning coefficient;
π·π, π·ππππ, π·π
πππ, π·πβ The capillary pressure (ππ), the predefined final (maximum)
capillary pressure reached after primary drainage (πππππ₯), the
predefined final (minimum) capillary pressure reached after
xvii
waterflooding (πππππ), and the threshold capillary pressure at
which the thin film collapses (ππβ);
PD Primary Drainage i.e. first oil charge in a water-filled and water-
wet system;
PD/WE The developed Primary Drainage/Wettability Evolution model,
where the wettability alteration evolves during Primary
Drainage (Chapter 5);
PL, SO Piston-like (PL) and snap-off (SO) pore-scale displacements;
PNM Pore network modelling;
PSD The distribution of pore-sizes within a network;
PV Pore volume of the porous medium (core/network);
πΈ The oil flow rate;
πΉπππ The inscribed radius of the cross-sectional shape of a pore;
πΊπ, πΊπ The water and oil saturations, respectively, in a network
(ππ = 1 β ππ€);
πΊππ, πΊππ The initial water and oil saturations, respectively, at which
waterflood starts following PD (πππ = 1 β ππ€π);
πΊππ The saturation of the residual oil following waterflood;
ππ The final time of the PD/WE simulation;
π· The limiting contact angle for which π€ = π€πππ₯ (π½ β [0,1]);
π The half-angle of the cross-sectional shape of a pore;
π(π), ππππ The adsorption level of polar compounds per unit area in a
given pore at time t (π€(t)) and the Langmuirβs maximum
adsorption level (π€πππ₯);
βππ»πΉ The transport modelβs time-step;
π½, π½ππ , π½πππ, π½π,ππ
and π½π,ππ
The oil-water contact angle (π), the primary drainage (fixed)
contact angle (πππ, Chapter 4), the maximum contact angle
reached following PD/WE (ππππ₯ , Chapter 5) and the water-wet
and oil-wet advancing contact angles, respectively (ππ,π€π€ and
ππ,ππ€);
Π, Πππππ Disjoining pressure (Π) and critical disjoining pressure at which
thin films collapse (Πππππ‘);
xviii
πππ Oil-water interfacial tension;
πππ, πποΏ½ΜοΏ½ Euler number (ππβ) and normalised Euler number (ππβΜ) of a
given phase πβ (oil/water) in a saturated network.
1
Chapter 1 : Introduction
Understanding the wettability state of a porous medium is essential for accurate
modelling of multi-phase flow processes in hydrocarbon reservoirs, as well as in aquifers
following contamination by non-aqueous phase liquids or injection of carbon-dioxide.
Specifically, in simulations of hydrocarbon reservoir behaviour, assumptions on the
wettability distribution strongly influence many petrophysical functions. This includes
the capillary pressure and relative permeability data, as well as the oil recovery
efficiency after waterflooding. The stakes become notably high when dealing with
carbonate rocks, as such formations host a significant amount of the remaining oil
reserves (Roehl and Choquette, 2012).
The pore-scale wettability of microporous carbonates has been unresolved. Indeed, the
micropores, which may be defined as pores <5 Β΅m in radius (Cantrell and Hagerty, 1999),
have usually been excluded from contributing to the flow process as they were often
supposed to be water-filled and water-wet. Nonetheless, the rise of imaging technology
in recent years helped visualise the pore-scale fluids distribution and wettability at the
micro and nanometre scale. In fact, many authors have detected oil in micropores using
high-resolution imaging (Al-Yousef et al., 1995, Clerke, 2009, Knackstedt et al., 2011,
Fung et al., 2011, Clerke et al., 2014, Dodd et al., 2014). This was an important step
forward to acknowledge the role that micropores might play in oil recovery processes,
as well as to recognise a possible oil-wet state in micropores. This was confirmed by
Marathe et al. (2012) who detected oil-wet micrite facets, and identified a face-selective
pattern of wettability alteration where curved surfaces are oil-wet, in contrast with
smooth water-wet surfaces. Nonetheless, questions arise about how oil migrates into
such tiny micropores without reaching huge capillary pressures. This might be linked to
a scenario of wettability alteration that has occurred over geological time.
In this work, we aim to model the pore-scale wettability alteration in complex multiscale
carbonate rocks. To achieve this, we use a quasi-static two-phase flow pore network
model that involves a wide range of pore shapes and includes a realistic thermodynamic
criterion for the formation and collapse of oil layers. Our main objectives consist of:
2
a) Reproducing the wettability alteration trends shown by high-resolution
imaging of microporous carbonates (Marathe et al., 2012).
b) Assessing the importance of the micropores' wettability on oil recovery.
c) Understanding the oil migration process into tiny micropores with
excessively high entry pressures.
In Chapter 2, we carry out in a first section an extensive review of the concept of
wettability in the literature. We first examine the different pore-scale mechanisms
responsible for the wettability alteration. Then, we describe the main factors that are
thought to affect wettability. Subsequently, we study how wettability is assessed from
pore to pore and how its pore-space distribution is characterized. Then, we examine the
significant effect of wettability on residual oil saturations. In the second section of this
chapter, we focus on the particular case of microporous carbonates, for which we begin
presenting their main structural features. It is followed by an analysis of the carbonatesβ
wettability characteristics and their oil recovery efficiency. Finally, we provide an
overview of the pore-scale modelling of multiscale carbonate rocks.
In Chapter 3, we briefly introduce the pore network modelling approach in general. Then
we present the quasi-static two-phase flow network modelling tool we use in this work
and describe its main features. Indeed, we present the variety of pore shapes modelled,
and the corresponding wide range of fluid configurations and multiple fluids
displacements. We then describe the commonly used workflow of primary drainage,
ageing and waterflooding, detailing the different pore-level displacement mechanisms
involved. We also introduce the Euler number as a measure of the connectivity of the
network and the fluid phases. Finally, we present the two topologically different pore
network models that are inputs to our simulations: a homogeneous Berea sandstone
network and a complex multiscale carbonate network.
In Chapter 4, we aim at meeting objectives a) and b). For this, based on the model
developed by Kovscek et al. (1993), we suggest a physically plausible wettability
distribution model following primary drainage, subsequent to ageing, that takes into
account both the pore size and shape. The resulting wettability distribution, referred to
as Altered-Wet, is compared to the standard wettability distributions that are either
exclusively correlated to pore size (Mixed-Wet Large and Mixed-Wet Small) or
distributed regardless of size (Fractionally-Wet). Then, we investigate the effect of the
3
wettability distribution on the petrophysical properties, especially on the residual oil
saturation, for both the Berea and carbonate networks.
In Chapter 5, we aim at meeting the final objective c). To achieve this, we suggest a novel
model of wettability alteration that occurs during primary drainage, based on the work
by Bennett et al. (2004). We thoroughly describe the model, consisting of a time-
dependent oil invasion coupled with a transport model for polar compounds, whose
adsorption triggers a wettability change. Then, for each of our input networks, we show
the results of primary drainage at the pore level, which we link to the reservoir scale.
Finally, we present the subsequent waterflood calculations and the resulting residual oil
saturation patterns, depending on whether an additional wettability alteration from
water-wet/intermediate-wet to oil-wet conditions during ageing was performed.
Finally, Chapter 6 consists of a summary of our main results and key findings. We further
recommend possible suggestions as future work.
4
Chapter 2 : Literature review
2.1 Wettability alteration
The wettability of a solid surface describes its degree of molecular affinity towards one
fluid (water/oil) rather than the other (Abdallah et al., 2007). Wettability is a surface
phenomenon, typically quantified using the equilibrium contact angle, π, measured
between the fluids at the interface, usually through the denser (water) phase, as shown
in Figure 2.1. Physically, the contact angle is a surface property describing the balance
between the surface and interfacial forces, unique for a given combination of a solid and
two pure fluids. It is computed through Youngβs equation:
ππ π = ππ π€ + πππ€πππ π (2.1)
where ππ π, ππ π€ and πππ€ are the surface-oil, surface-water and oil-water interfacial
tensions, respectively. However, for a real crude oil/brine/rock (COBR) system, the
wettability cannot be characterised by a single contact angle but rather by numerous
contact angles at the different three-phase contact points zones (Hirasaki, 1991). For
instance, two different contact angles can be depicted, the advancing and receding
contact angles, depending on whether the water is the displacing or the displaced phase,
respectively. This phenomenon, the so-called contact angle hysteresis, is mainly a result
of surface chemical heterogeneity and roughness. The advancing contact angle is usually
higher than the receding contact angle, and the gap between the two values grows with
increased roughness (Morrow, 1975).
Figure 2.1: Wettability illustrated using a pure water drop at equilibrium on a smooth
solid surface, surrounded by pure oil. If π < 90Β°, the surface is water-wet and the oil
drop spreads on the surface. Otherwise, if π > 90Β°, the surface is oil-wet and the oil
drop forms a bead, minimizing the contact with the solid.
5
2.1.1 Pore-scale wettability alteration mechanisms
Reservoir rocks are thought to have been initially filled by formation water for a long
geological time thus having an initial strong water-wet state before oil migration
(Marzouk, 1999, Hamon, 2000, Abdallah et al., 2007). The pore-scale wettability
alteration of reservoir rocks has often been associated with asphaltenes, which have
been cited as being major wetting alteration agents. These are high molecular weight
polar aggregates occurring in many crude oils, coated by lower molecular weight
molecules called resins. Asphaltenes are characterised by a high surface activity.
However, their large average size of ~0.6 ππ is thought to prevent them from
accessing pores smaller than their own size (Al-Yousef et al., 1995). In addition,
asphaltenes have the characteristic of being highly insoluble in water, i.e. hydrophobic.
This prevents them from penetrating thin protective water films that occur on water-
wet surfaces (Hirasaki, 1991) and directly contacting the pore walls.
Using βadhesion testsβ on smooth silicate surfaces, Buckley and Liu (1998) identified
four mechanisms that are jointly responsible for the adsorption of such polar
components from the crude oil onto the surface: polar interactions, surface
precipitation, acid/base interactions and ion-binding:
Polar interactions occur between polar compounds in the oil and polar sites on
the surface. They only contribute when the oil is directly in contact with the
surface i.e. in the absence of a protective water film, thus are unlikely to take
part in initially aqueous systems. This mechanism leads to moderate wettability
alteration to intermediate-wet conditions.
The surface precipitation of asphaltenes may take place in the case where the
crude oilβs solvency for its asphaltenes is low. This leads to a relatively high
wettability alteration to oil-wet conditions.
In the presence of a water film, Coulombic acid/base interactions occur between
ionized acidic and basic components at electrically charged oil/water and
solid/water interfaces. These interactions control the surface charge at the
interfaces, which in turn affects both the water-film stability and surface
adsorption. The resulting contact angles depend on both the water composition
(pH and ionic properties) and oil properties (acid and base numbers).
6
Ion binding interactions may prevail over the acid/base interactions in the
presence of divalent and/or multivalent ions in the brine (e.g. Ca2+). These ions
bind at the oil/water and solid/water interfaces and influence the contact angle.
As for the acid/base interactions, the extent of the wettability alteration depends
mainly on the oil and brine compositions.
Kovscek et al. (1993) established a pore-level scenario to describe the wettability
alteration of initially water-wet rocks following ageing in crude oil. It is based on the
existence of the thin films that coat the pore wall and preserve the rockβs initial
wettability state. At equilibrium, the water film in a given pore is stabilised by thin film
forces; the related pressure is the so called disjoining pressure, Π. Three major factors
contribute to the disjoining pressure: Electrostatic interactions, Van der Waals
interactions and Hydration forces (Hirasaki, 1991). As shown in the illustrative disjoining
pressure isotherm (Figure 2.2), Π depends on the thickness of the water film. As the
capillary pressure ππ = ππ β ππ€ i.e. the difference in pressure at the oil/water interface,
rises during oil invasion, the film gets thinner. The film first gains in stability since Π
rises, until ππ reaches a threshold capillary pressure, ππβ:
ππ
β = Πππππ‘ + πππ€π (2.2)
The threshold capillary pressure is an intrinsic property of the pore related to the oil-
water interfacial tension, πππ€, the curvature of the pore wall, π, and the critical
disjoining pressure at which the film collapses, Πππππ‘. The latter depends on the fluid
system and mineralogy of the rock surface. In fact, when ππ > ππβ , π. π. Π < Πππππ‘, the
film becomes unstable and breaks down to a molecularly thin one. Subsequently, polar
compounds from the oil, mainly asphaltenes, may then irreversibly adsorb onto the
surface, hence rendering it oil-wet.
Figure 2.2: Evolution of thin film stability during oil invasion, illustrated on an example
of disjoining pressure isotherm Π(π), after Hirasaki (1991).
7
Kovscek et al. (1993) incorporated their wettability alteration mechanism in a capillary
bundle model with star-shaped pore cross sections. The resulting capillary pressure
curves and residual oil showed reasonable qualitative agreement with experimental
results. Frette et al. (2009) also included this scenario in a bundle-of-tubes theoretical
model with 2D realistic pore cross sections derived from high resolution SEM (Scanning
Electron Microscopy) images, for which they examined the relationship between
collapsed films fraction and capillary pressure. It has also been implemented in 3D
network models (Blunt, 1997, Blunt, 1998, Oren et al., 1998, Jackson et al., 2003). To
allow wettability changes to occur, these models employed angular pore cross sections,
and a simple parametric model for the water film collapse.
A complementary theory suggests that the thin water films may initially be destabilised
by the adsorption of polar components with smaller molecular weight that are present
in the crude oil (van Duin and Larter, 2001, Bennett et al., 2004, Bennett et al., 2007).
Indeed, crude oils are usually rich in smaller polar non-hydrocarbon compounds, for
instance the aromatic oxygen compounds such as alkylphenols (e.g. phenol πΆ0 β πΆ3)
(Taylor et al., 1997, Lucach et al., 2002). Alkylphenols are characterized by their high
solubility in water (hydrophilic, unlike asphaltenes) and high surface activity. Indeed,
Huang et al. (1996) induced wetting changes in a laminated rock using a βsynthetic crude
oilβ containing a number of candidate smaller polar molecules such as cresols, phenols,
carbazoles, etc. van Duin and Larter (2001) used molecular dynamics simulations to
suggest a wettability alteration process involving these small water-soluble polar non-
hydrocarbon compounds. First, they penetrate the water films that initially coat water-
wet mineral surfaces. They then rapidly adsorb onto the surface and render it more
hydrophobic. As a result, the water film is disrupted and the disjoining pressure is
reduced thus making the film prone to collapse at the existing local oil/water capillary
pressure. This water film collapse then allows direct contact with the surface by heavier
compounds, such as asphaltenes. As a result, the surface is rendered oil-wet with
contact angles reaching up to 180Β°.
Bennett et al. (2004) confirmed this wettability alteration process using a core-flood
experiment on a sandstone, where they observed an early rapid wettability alteration
that occurred during primary drainage involving alkylphenols. Indeed, these small polar
species were absent from the eluted oils at the end of the experiment (βp-cresolβ in
8
Figure 2.3(a)). The non-appearance of polar compounds is mainly due to their high
interaction with the surface, which decreased across the length of the core (Figure
2.3(b)). This resulted in a significant wettability alteration, preferentially near the inlet,
as confirmed by an ESEM (Environmental SEM) examination of the core after wetting
alteration. In fact, while water spread as continuous sheets near the outlet, supporting
a water-wet behaviour (Figure 2.3(c)), it appeared as discrete droplets on surfaces near
the inlet, supporting rather oil-wet conditions (Figure 2.3(d)). Additionally, the authors
pointed out the relative speed of the process, which may happen in a reservoir over a
time-scale from days to months. Thus, the timescale of wetting alteration in such a
mechanism would depend on the slowest step in the process from diffusion of the polar
organics into the water films, the adsorption of these smaller polar molecules to the
rock surface, the lowering of the disjoining pressure of the water film and its subsequent
collapse and finally the adhesion of larger polar compounds such as asphaltenes onto
the rock surface, which is unknown.
(a) (b)
(c) (d)
Figure 2.3: (a) Plot of the normalized concentrations of polar compounds in the
produced oil (mobile) function of the elution time; (b) Plot of the concentration of βp-
cresolβ in the core extract oil (immobile) relative to the distance along the core; ESEM
images showing water spreading trends from (c) the outlet and (d) inlet of the core
(Bennett et al., 2004).
9
The scenario suggested by Bennett et al. (2004) is compatible with the findings by Graue
et al. (2002) who performed core-flood experiments on chalk (carbonate) rocks. They
observed that the wettability alteration process was more efficient by continuous
injection of crude oil during ageing, as opposed to stopping the oil invasion. Additionally,
the authors pointed out the important increase in wettability alteration efficiency with
the core length, as the continuously flushed oil was resupplying its surface-active polar
compounds, thus speeding up the ageing process.
2.1.2 Factors affecting wettability
A multitude of factors are thought to affect wettability, which is a property of the whole
COBR system. These widely include the rockβs intrinsic properties, the brine and crude
oil properties as well as the ageing history. Nonetheless, the importance of each of the
factors cited above is not agreed upon in the literature as it depends on the specific
system studied.
The rockβs intrinsic properties
Some intrinsic properties of the rock, such as the geometry of the pores (size, roughness,
etc.) and surface mineralogy, are thought to affect wettability. Indeed, Fassi-Fihri et al.
(1995) observed that wettability in carbonates was mostly affected by pore geometry,
while mineralogy was found to be the major factor controlling wettability in sandstones.
On the other hand, Hamon (2000) examined a large dataset of samples from a sandstone
hydrocarbon reservoir and found no evidence that pore mineralogy has any direct
impact on wettability. He rather observed that horizontal wettability variations across
the reservoir (at constant capillary pressures) are dependent on the pore geometry
through the sampleβs permeability, with more oil-wet conditions for higher
permeabilities.
The brine characteristics
The brine characteristics, mainly identified as its ionic composition and pH, may play a
role in wettability alteration. Indeed, Mahani et al. (2015) demonstrated that for
sandstones, wettability in the presence of clay is mostly dependent on the brineβs ionic
composition.
10
The crude oil composition
Buckley and Liu (1998) focused on the effect of the crude oil composition on the
wettability by comparing crude oils with strikingly different compositions. They
identified three chemical characteristics that determine the potential of a crude oil to
alter the surface wettability: its acid number, its base number and its API gravity (or
density), which measures its solvency for its polar components (e.g. asphaltenes).
Interestingly, they showed that the solvent environment to which the asphaltenes
belong is more important to their potential wetting performance than their actual
amount. In fact, the poorer the solvent quality of a crude oil to its asphaltenes, the
stronger would be the resulting oil-wet conditions.
Besides, Idowu et al. (2015) examined using high-resolution imaging techniques two
(cleaned) mini-plugs of an outcrop and a reservoir sandstone following ageing in crude
oil. They showed that the samples ended up in uniformly water-wet and oil-wet
conditions, respectively, even though they were characterized by quasi-identical
mineralogies (dominated by quartz and plagioclase). They suggested that the observed
deviations in wetting behaviour were mainly due to the difference in the crude oils they
utilized for each sample.
The saturation and ageing history
The wettability of reservoir rocks depends strongly on their original structural position
(Marzouk, 1999, Hamon, 2000, Abdallah et al., 2007, Okasha et al., 2007). In fact,
wettability generally varies vertically in the reservoir with height above the oil-water
contact (OWC) i.e. the highest level in the oil column at which the oil saturation is zero.
Indeed, the higher the sample in the oil column, the more oil-wet it is likely to be. This
results in the establishment of a mixed wettability state in the transition zone (as
illustrated in Figure 2.4). This wetting behaviour can be attributed to three major factors:
The capillary pressure being higher at the top, which results in thin films coating
the pore walls becoming less stable (Hamon, 2000) (see Section 2.1.1).
Oil accumulated from top to bottom, hence oil contacted the upper surfaces for
a longer time (Marzouk, 1999).
A higher number of pores are invaded at the top, thus the overall sampleβs
affinity to oil is more likely to increase (Jackson et al., 2003).
11
Figure 2.4: Structural position effect on fluid saturations and wettability within a typical
reservoir. In the water and oil zones, all pores are water-filled & water-wet and oil-
filled & oil-wet, respectively. In the transition zone, the smallest and largest pores are
water-filled & water-wet and oil-filled & oil-wet, respectively. However, some
intermediate-sized pores may be oil-filled & water-wet, as the thin water film may
resist the moderate capillary pressure exerted.
The ageing time and temperature are also thought to affect wettability. As for the ageing
time, if any correlation exists, it tends to be positive with respect to the oil-wetness
(Buckley and Liu, 1998, Kowalewski et al., 2003). Rock samples are typically brought
from an initial water-wet to an oil-wet state by aging in crude oil in labs for a duration
ranging from several days to a few weeks. Kowalewski et al. (2003) observed that ageing
sandstone samples in crude oil at 75 β beyond 10 days did not induce any further
changes to their wettability. However, the relationship between the ageing temperature
and extent of wettability change is still debated in the literature. While Buckley and Liu
(1998) observed an increase in the measured contact angles with temperature on a glass
surface; the opposite effect was reported by Lichaa et al. (1993) on calcite surfaces from
three different carbonate samples.
2.1.3 Assessing wettability
No universal method is currently available to accurately evaluate the wettability of a real
COBR system, which cannot be described by a unique contact angle. Instead, both
quantitative and qualitative methods were designed to provide a description of
wettability as close as possible to the real one.
12
a) Quantitative methods
On the one hand, direct quantification of wettability can be done through contact angle
measurements, traditionally from pure mineral surfaces, or more recently using high-
resolution imaging techniques. On the other hand, indirect methods use capillary
pressure curves to infer an average wettability state of the core.
Contact angle measurements
Contact angles can be directly evaluated from mineral surfaces. For this, several
methods have been developed, thoroughly reviewed by Yuan and Lee (2013). These
include the commonly used sessile drop, the captive bubble and the Wilhelmy balance
methods. The main shortcoming of these techniques is that they require a rigid, smooth,
homogeneous surface interacting with a pure fluid. Hence, they fail to capture the full
heterogeneity of a real system.
Important progress in imaging led to the development of advanced techniques such as
the Drop Shape Analysis (DSA) method and its improvement the Axisymmetric Drop
Shape Analysis (ADSA) method (Yuan and Lee, 2013). These techniques evaluate
the contact angle by analysing the shape of a drop and fitting it into a theoretical profile.
They are undergoing continuous improvement, leading to ever more accurate
wettability measurements and a much enhanced reproducibility.
Andrew et al. (2014) suggested a new method to directly measure the contact angles
from micro-CT (micro X-ray computed Tomography) images. It is based on manual
tracings of vectors tangential to the non-wetting/wetting interface and the solid surface,
as shown in Figure 2.5. They applied this technique on a CO2-brine-carbonate system
and obtained a distribution of contact angles centred around 45Β° i.e. the system was
weakly water-wet. However, the method requires hundreds of manual tracings.
Moreover, at the microscale, thin water films (nanometre-scale) are not resolvable,
hence the three-phase contact line -if existent- may not be accurately identified.
13
Figure 2.5: An example of a measured contact angle (53Β°); it is the complement of the
traced angle (pink arc, 127Β°) measured through the CO2 (black, less dense phase)
(Andrew et al., 2014).
Amott / USBM
To evaluate the average wettability of reservoir core samples, indirect methods
requiring capillary pressure data are used, based on displacement analysis, namely the
Amott-Harvey index (IAH) (Amott, 1959) and United States Bureau of Mines index (IUSBM)
(Donaldson et al., 1969). Figure 2.6 gives a brief overview on how these are calculated.
Practically, both indices range between β1 for strongly oil-wet to 1 for strongly water-
wet conditions. They can be calculated from a single combined experiment (Sharma and
Wunderlich, 1987). Note that neither of the tests is based on a well-established theory,
they are rather empirical. Moreover, even though they have been of interchangeable
usage in the literature, the link between them from a theoretical and practical point of
view remains ambiguous. Indeed, while the Amott index is controlled by spontaneous
displacement, the USBM index mainly depends on the energy associated with forced
displacement. Dixit et al. (2000) used both analytical computations and the pore
network modelling approach to examine the relationship between the two indices. They
demonstrated that indeed a broad correlation exists between them, further explained
in Section 2.1.4, which can provide further interpretation of some experimental results.
Moreover, they showed that both indices depend on the network characteristics (e.g.
pore size distribution, network connectivity). This was consistent with experimental
results conducted later by Hamon (2000) who clearly states that these measures are not
a pure reflection of the intrinsic wettability of the rock.
14
Figure 2.6: Calculation of the Amott indices to water,πΌπ€, and oil, πΌπ, and their
combination through the Amott-Harvey index πΌπ΄π», as well as the USBM index, πΌπππ΅π.
These depend on the water saturations, ππ, at the end of the flood π ( π = 1 β primary
drainage, 2 - spontaneous imbibition, 3 β forced imbibition, 4 β spontaneous drainage,
5 - forced drainage) (Dixit et al., 2000).
b) Qualitative methods
Qualitatively, imaging techniques are able to reveal wettability distributions in rocks at
the level of individual pores. For instance, Cryo-SEM (cryogenic SEM) was used to infer
the local wettability from the visualised fluid distributions at the microscale (Fassi-Fihri
et al., 1995, Al-Yousef et al., 1995, Kowalewski et al., 2003). In fact, Fassi-Fihri et al.
(1995) confirmed the heterogeneity of wettability at the pore scale. They were also able
to observe wetting films, but only those thicker than the imaging resolution of 0.1ππ.
The Atomic Force Microscopy (AFM) technique (Lord and Buckley, 2002, Morrow and
Buckley, 2006) can be used to visualise nano to micron-scale topographic variations of
mineral surfaces following contact with crude oil. Lord and Buckley (2002) used AFM to
describe the properties of the adsorbed film on smooth mica surfaces exposed to crude
oil. They were able to resolve the micron-scale asphaltene aggregates deposited on the
surface, cemented to it through an adsorbed film. The latter consists of other polar
compounds from the crude oil, and is of about 15 nm thickness. Morrow and Buckley
(2006) observed similar trends when ageing calcite surfaces (Figure 2.7).
15
(a) (b)
Figure 2.7: 10Β΅m x10Β΅m AFM image (scanned under water) of a calcite surface aged in
crude oil for (a) 2 days and (b) 21 days. Note the presence of a continuous adsorbate
layer covering the surface in (a), which gets thicker and more stable by increasing the
ageing time in (b) with the presence of large particles with irregular morphologies
(asphaltenes aggregates) (Morrow and Buckley, 2006).
In the last couple of years, significant advances in imaging technologies helped visualize
details of the local wettability alteration at higher resolution. Recently, FESEM (Field-
Emission SEM) was used to image the asphaltenes deposition i.e. the wettability
βfootprintβ, on microparticle faces in dry conditions, which are revealed after the
removal of the fluids (oil and brine) with mild solvents (Knackstedt et al., 2011, Marathe
et al., 2012, Dodd et al., 2014, Idowu et al., 2015).
Schmatz et al. (2015) used a combination of the cryo-BIB (broad ion-beam), SEM, and
EDX (energy-dispersive X-ray) analysis techniques to image details of the pore-scale
fluid-fluid-solid contacts at the nanoscale in a mixed-wet sandstone. All the observed
oil-water-rock contacts correspond to one (or are a combination) of three distinct
configurations (Figure 2.8). Moreover, they were able to directly measure contact angles
using serial sectioning technique when the three-phase contact line was distinguishable
(category (a) in Figure 2.8). However, this was far from being straightforward due to the
complex local geometries at the nanoscale.
16
Figure 2.8: The observed oil-water-rock contacts which fall in three different categories:
(a) a conventional three-phase contact line; (b) a thin water film keeping the solid
surface from direct contact with oil; and (c) a local oil-rock pinning at geometrical and
chemical heterogeneities (asperities) (Schmatz et al., 2015).
Idowu et al. (2015) proposed to integrate the information from the local surface
mineralogy visualised by high-resolution QEMSCAN (Quantitative Evaluation of Minerals
by SEM) with the local wettability alteration displayed by FESEM. Their method suggests
that this gathered βmineral-dependentβ wettability information would then be
incorporated into a pore network model. However, the mineralogy information they
collected did not help in understanding the wettability distributions, since the two
samples they chose possessed quite similar mineralogies and showed uniformly water-
wet and oil-wet conditions, respectively. Moreover, it is unclear how the wettability
information would be included on a pore by pore basis in a pore network model. The
main challenges to address are the difficulty to capture the full heterogeneity of a mixed-
wet system, as well as the absence of information on the contact angles.
2.1.4 Wettability distributions
Oil recovery in a porous medium is not only influenced by its degree of wetting change
but also by the distribution of wettability among the pores. McDougall and Sorbie (1995)
proposed a classification system from pore-to-pore that consists of uniform and non-
uniform wettability distributions:
Uniform wettability: homogeneous wetting behaviour throughout the rock
where all the pores are either uniformly water-wet, intermediate-wet (i.e.
neutral-wet) or oil-wet. Many cut-off contact angles are available in the literature
to classify the intermediate-wet state, for instance between 75 and 105Β°
(Anderson, 1986).
Non-uniform wettability: wettability distributed non-uniformly from pore-to-
pore, with coexisting fractions of water-wet and oil-wet pores in the rock. This
17
category is classified into sub-categories, depending on the correlation between
pore wettability and size:
o Mixed-Wet Large (MWL): large pores are oil-wet, small pores are water-
wet;
o Mixed-Wet Small (MWS): small pores are oil-wet, large pores are water-
wet;
o Fractionally-Wet (FW): wettability randomly distributed, uncorrelated to
pore size.
McDougall and Sorbie (1995) evoked as well the possible existence of non-uniform
wettability within the same pore, with the presence of oil-wet and water-wet pore walls.
However, they did not examine this further. The described classification system is
summarised in Figure 2.9 and will be employed throughout the thesis.
Figure 2.9: Wettability classification system, after Ryazanov (2012).
The consequences of the MWL, MWS and FW wettability distributions on the observed
wettability indices (both IAH and IUSBM) and their relationship were predicted by Dixit et
al. (2000) using analytical calculations and pore network modelling. Under some
simplifying assumptions, this method predicts that the FW distribution of a slightly
water-wet nature coincides with the IAH =IUSBM line. On the other hand, the MWL and
MWS are predicted to reside above and below the IAH =IUSBM line, respectively, as shown
in Figure 2.10, with the majority of the data points lying in the MWL region.
18
Figure 2.10: Relationship between IAH and IUSBM indices that were measured from the
same core sample by different authors. The MWL, FW and MWS lines are derived
analytically (Dixit et al., 2000).
Actually, the MWL distribution, firstly introduced by Salathiel (1973), is thought to be
the most widely occurring wettability distribution. The simple reason behind that is that
the oil, naturally present in the larger pores, would preferentially alter their wettability.
Hamon (2000) observed trends from sandstone reservoirs compatible with the MWL
distribution. Indeed, wettability increased horizontally (at fixed height in the oil column)
with permeability, which can be translated into larger pore spaces.
Nonetheless, there is obvious evidence from the literature of the existence of all
wettability distributions in real reservoir rocks. RueslΓ₯tten et al. (1994) classified using
the IAH /IUSBM correlation two reservoir plugs as being MWS. A thorough examination of
these samples showed the presence of oil in the smallest pores, which makes the least
conventional MWS condition plausible. Additionally, Skauge et al. (2007) proved that
MWL, MWS and FW distributions are all theoretically compatible with the scenario
suggested by Kovscek et al. (1993) involving the disjoining pressure concept (see section
2.1.1). The former demonstrate that the type of wettability distribution may be fully
determined by the stability of the thin films, which in turn is controlled by the curvatures
of the pore-shapes. In fact, MWL, MWS and FW distributions are related to mono-
mineral rock materials consisting of concave (positive curvature), convex (negative
curvature) and flat (zero curvature) pore-shapes, respectively. Additionally, they
19
examined a large dataset of wettability indices from intermediate-wet sandstone
reservoirs, each of them falling into one of the MWL, MWS and FW categories according
to Dixit et al. (2000)βs correlation. The predicted wettability distributions were further
confirmed by examining the related SEM/ESEM images.
Although helpful to study homogeneous cases, these theoretical wettability
distributions may be insufficient to describe complex wettability distributions that may
arise in heterogeneous pore spaces, especially in microporous carbonates (refer to
Section 2.2.2).
2.1.5 Effect of wettability on the residual oil saturation
Residual oil saturations after waterflooding in oil-bearing reservoirs may range from less
than 1% to more than 40% (Morrow, 1990, Skauge and Ottesen, 2002). An essential
feature to understanding this large scatter in oil recovery efficiencies is thought to be
the wettability. Actually, the role of wettability on oil recovery is crucial as it is a property
that controls the fluids displacement and trapping mechanisms, hence eventually
influences the final oil distribution and residual saturation.
a) Experimental work
Salathiel (1973), for the first time, related an unexpectedly high oil recovery to a mixed-
wet (large) system in a sandstone reservoir. He explained this behaviour by the
persistence of oil-wetting films in the surfaces of altered wettability during
waterflooding. These films would maintain the oil phase connectivity and form a
continuous pathway for oil to flow to remarkably low saturations (lower than 10%).
Jadhunandan and Morrow (1995) conducted waterflooding experiments on Berea
sandstone samples by injecting up to 20 pore volumes (PV) of water. They found that
the residual oil saturations decreased by inducing wettability changes from strongly
water-wet to a minimum at very weakly water-wet to neutral-wet conditions (Figure
2.11). Additionally, the recovery efficiency was steadily improving as more water was
injected, as shown in Figure 2.11.
20
Figure 2.11: Residual oil saturation as a function of the Amott-Harvey index for three
different crude oils and different PV of water injected (Jadhunandan and Morrow,
1995).
Skauge and Ottesen (2002) published oil recovery results from a wide dataset of around
30 sandstone reservoirs shown in Figure 2.12. Although they were overall consistent
with Jadhunandan and Morrow (1995)βs findings, they observed a large scatter in the
residual saturation data, especially at neutral-wet conditions. This reflects the inability
of the used wettability indices to cover the full extent of wettability alteration, and/or is
due to the difference in the experimental materials (COBR systems) and procedures.
21
Figure 2.12: Residual oil saturation as a function of the Amott-Harvey index from 30
sandstone reservoirs (Skauge and Ottesen, 2002).
Skauge et al. (2007) extended the previous work using a wider dataset and further
studied the effect of the different wettability distributions (MWL, MWS and FW) on the
oil recovery from sandstone reservoirs. They observed a trend where the MWL, FW and
MWS systems generally exhibited increasing residual oil saturations, in order.
b) Pore-scale simulations
Many authors have studied the effect of wettability on oil recovery using capillary-
dominated pore network modelling (PNM) (McDougall and Sorbie, 1995, Blunt, 1997,
Blunt, 1998, Dixit et al., 1999, Zhao et al., 2010, Ryazanov et al., 2014). In these pore-
scale simulations, pore-level displacement mechanisms (piston-like, snap-off, oil layer
formation, etc.), which occur depending on the governing wettability state, control the
fluids distributions and trapping behaviour. These will be detailed in Chapter 3.
First, there is a need to define the residual oil saturation, πππ, from a simulation point of
view. Ryazanov et al. (2014) distinguished between four possible definitions of πππ:
The waterflood is stopped at a predefined πππ. This value is not a prediction of
the residual oil, but should be rather referred to as the βremainingβ oil
saturation.
The waterflood is stopped when a predefined final (minimum) capillary
pressure, πππππ is reached. This should be referred to as πππ at ππ
πππ.
The waterflood is carried out until no more pore-level invasions are possible
under the capillary-dominated regime. Since the capillary pressure is lowered to
22
infinite levels (more specifically to the lowest available entry pressure), this
should be referred to as the βultimate recoveryβ, or πππ at πππππ β ββ. This is
considered to be a theoretical limit, probably impractical in core-flood
experiments.
The waterflood is stopped when a predefined number π of pore volumes (PV) of
water is injected. This threshold is obtained from the network simulation by
integrating the resulting relative permeabilities into a one-dimensional
continuum scale Buckley-Leverett model (Dake, 1983). This definition of the
residual oil, referred to as πππ at πPV, has been used extensively in the literature
in order to compare with experimental data (McDougall and Sorbie, 1995, Dixit
et al., 1999, Valvatne and Blunt, 2004, Zhao et al., 2010, Ryazanov et al., 2014).
McDougall and Sorbie (1995) examined the effect of both wettability distribution (FW
and MWL only) and oil-wet fraction on oil recovery by simply using a regular 3D cubic
network constituted of pores with circular cross-section. They concluded that for both
FW and MWL, the lowest residual oil saturation following up to 20PV of water injected
was reached when approximately half of the pores are oil-wet. McDougall et al. (1997)
and Dixit et al. (1999) extended the latter work by introducing a regime-based
framework for classifying and interpreting oil-recovery experiments depending on the
underlying wettability properties (contact angles and oil-wet fraction).
Blunt (1998) used a regular cubic network with square cross-sectional pores and a
parametric model to model the pore-scale wettability scenario suggested by Kovscek et
al. (1993). He demonstrated the complexity and non-monotonicity of the relationship
between the oil recovery and wettability, characterized by the oil-wet fraction and
contact angle. In fact, as shown in Figure 2.13(a), he identified four governing regimes
depending on the range of contact angles:
i. Strongly water-wet: The competition between snap-off and piston-like dictates
the amount of residual oil. Note that snap-off is considered as being the major
trapping mechanism.
ii. Weakly water-wet: Snap-off is inhibited, and water invades throats in favour of
nodes, hence this regime leads to a low trapping and a high recovery.
23
iii. Weakly oil-wet: Water preferably invades large throats and pores, hence it tends
to surround the wetting oil phase. As a result, this regime is characterized by a
high trapping, leading to a poor recovery.
iv. Strongly oil-wet: Oil layers are formed in oil-wet pores and enhance the recovery.
The ultimate residual oil saturations reach very low values, but only following
high PV of water injected.
Additionally, as shown in Figure 2.13(b), Blunt (1998) observed that only at an oil-wet
fraction of around 0.55, when a connected pathway of oil-wet pores is formed
throughout the network, that the oil recovery starts improving. The recovery then gets
better by means of slow displacements through connecting oil layers.
(a) (b)
Figure 2.13: Ultimate residual oil saturation as a function of contact angle (left) and oil-
wet fraction (right) (Blunt, 1998). Note that βSnap-offβ, βNucleated wettingβ, βNon-
wetting advanceβ and βOil layersβ correspond to the regimes i., ii., iii. and iv. as
described in the text, respectively.
Oren et al. (1998) reconstructed a geologically realistic sandstone network based on
thin-section data analysis. Their model has been extensively used in attempts to
reproduce experimental trends. Jackson et al. (2003), Valvatne and Blunt (2004) and
Zhao et al. (2010) aimed at reproducing the experiment conducted by Jadhunandan and
Morrow (1995) by calculating πππ at 3PV. Although the simulations are in fairly good
agreement with the experiment (see Figure 2.14), the maximum oil recovery is slightly
moved to the more oil-wet regime (lower Amott-Harvey index). Note that the wettability
was used as a tuning parameter to match the experiment: the oil-wet fraction and the
distribution of oil-wet contact angles were varied to reproduce the Amott water and oil
24
indices, respectively. Furthermore, Zhao et al. (2010) examined the impact of the
distribution of contact angles and the oil-wet fraction on the residual oil saturation after
3PV of water injected using several networks with different pore systems derived from
micro-CT images of real rocks. They concluded that for a uniformly-wet state, the
residual oil saturations decrease with higher contact angles until it reaches a plateau for
slightly oil-wet conditions at around 100Β° where the best oil recovery is obtained.
However, for a mixed-wet state, the fraction of oil-wet pores has a more significant
impact on oil recovery than the contact angle. In fact, a large fraction of oil-wet pores
leads to the lowest residual oil saturation. Additionally, they observed that the effect of
the pore structure on the relationship between wettability and oil recovery is weak.
Figure 2.14: Comparison between PNM simulation and experimental data from
Jadhunandan and Morrow (1995) of the oil recovery (the Fraction of Oil In Place) as a
function of wettability (Amott-Harvey index) following a 3PV injection of water (Zhao et
al., 2010).
Ryazanov et al. (2014) used the pore-network modelling tool described in Chapter 3 to
obtain a better match of the Jadhunandan and Morrow (1995) experiment compared to
previous studies, including at different PV throughput (Figure 2.15). They also visualised
the 3D structure of the residual oil in mixed-wet systems as a function of wettability.
They were able to observe clear variations in the nature of the residual oil, depending
on the systemβs wettability condition. On the one hand, for moderately water-wet
conditions, the residual oil is detected in the bulk space of pores. Additionally, its
saturation is high at relatively small numbers of PV of water injected. On the other hand,
for moderately oil-wet conditions, the ultimate residual oil resides in a large number of
25
oil layers with relatively low volume, and in few bulk spaces. Moreover, πππ slowly
increases with higher PV injected since the flow is mainly controlled by the connecting
oil layers.
Figure 2.15: Comparison between PNM simulation and experimental data from
Jadhunandan and Morrow (1995) of the recovery factor (RF) as a function of
wettability (Amott-Harvey index) at breakthrough (BT), and following 3PV and 20PV
injection of water. The infinite (inf) PV case was also included on each figure (Ryazanov
et al., 2014).
2.2 Carbonates and microporosity
Around half of the worldβs remaining hydrocarbon reserves lie in carbonate reservoirs
(Roehl and Choquette, 2012). The carbonatesβ complexity resides in their
heterogeneous pore space, with pore sizes spanning multiple length-scales. Micropores
are particularly interesting to examine as they may dominate the connected pore system
in many carbonate reservoirs, accounting for up to 100% of the total porosity (Cantrell
and Hagerty, 1999). Among many definitions of micropores available in the literature,
we use the definition suggested by Cantrell and Hagerty (1999) as pores <5 Β΅m in radius.
Nonetheless, microporosity has been poorly understood and its contribution to flow has
often been neglected, since micropores are usually assumed to contain irreducible
water due to their small size and the resulting high capillary pressures required for the
oil to enter them, hence they are naturally assumed to be water-wet. However, the
presence of hydrocarbons in micropores was identified (Al-Yousef et al., 1995, Clerke,
2009, Knackstedt et al., 2011, Fung et al., 2011, Clerke et al., 2014, Dodd et al., 2014),
which may result in a potential oil-wet state. This has driven research to analyse the
wettability of micropores, thought to be the key to maximise oil recovery from
microporous carbonates, which is the main topic of this thesis.
26
2.2.1 Composition and structure
Despite a significant variability in pore structure, carbonates possess a simple
mineralogy compared to sandstones, being mainly dominated by calcite, and to a lesser
extent by dolomite. These minerals typically have a slightly basic nature and exhibit a
positive surface charge (Lichaa et al., 1993, Morrow and Buckley, 2006). Morrow and
Buckley (2006) state that this is generally true when the pH of the brine in contact with
the surface is around neutral.
Microporosity in carbonates often occurs within micrite (microcrystalline calcite)
crystals of rhombic shape (Figure 2.16(a)). It is thought to have formed due to diagenesis
i.e. carbonate dissolution, transport and precipitation (cementation) mechanisms.
Diagenetic processes require an aqueous medium, thus are not favoured in presence of
the oil and predominant oil-wet conditions (Cox et al., 2010, Heasley et al., 2000).
Cantrell and Hagerty (1999) suggested that several diagenetic mechanisms account for
the formation of microporosity. Moreover, they observed that micropores may exist in
many types, namely as (a) microporous grains, (b) microporous matrix, (c) microporous
fibrous to bladed cements, and (d) microporous equant cements; as illustrated in Figure
2.17. Whatever the type, a magnified view shows an identical sponge-like microporous
structure, not only well inter-connected but also connecting up the meso/macro-
porosity (Figure 2.16(b)). Individually, the micropores usually exhibit a distinctive
morphology consisting of flat and platy shape. The micrite crystals sizes, shapes and
packing pattern influence the micropores morphology (Harland et al., 2015).
Additionally, Cantrell and Hagerty (1999) state that the wide variations in the
microporosity types and abundance (ranging between 0 and 100% of the total porosity),
are mainly due to the depositional texture (in particular the presence or absence of
mud).
27
(a) (b)
Figure 2.16: SEM images from microporous carbonates (a) displaying the rhomboidal
micrite crystals and (b) showing the structure of the microporous network and its
connection to the larger mesopore (using epoxy resin cast) (Harland et al., 2015).
(a) (b)
(c) (d)
Figure 2.17: Illustration of the four different microporosity types observed by Cantrell
and Hagerty (1999): (a) microporous grains, where the fully micritised grains seem like
sponges; (b) microporous matrix, consisting of a network of connected micropores; (c)
microporous fibrous to bladed cements, where micropores are found between cement
blades; and (d) microporous equant cements, with the appearance of micropores
between cement crystals.
28
The complex heterogeneity of carbonate pores has led many authors to arrange them
into pore classes according to their structure, porosity and permeability. Naturally, the
more heterogeneous carbonates would contain more pore classes. Several classification
systems exist in the literature, constituted by tens of pore types that can be visually
identified from rock images (Choquette and Pray, 1970, Lucia, 1983, LΓΈnΓΈy, 2006). It is
well established that flow properties vary significantly with pore classes (Skauge et al.,
2006). Additionally, this categorisation enhances our understanding of the relationship
between some of the petrophysical properties for each class (e.g. porosity, permeability,
etc.) and the resulting residual oil saturations after waterflooding (Pourmohammadi et
al., 2008).
2.2.2 Wettability
Earlier in section 2.1, we described more generally the wettability for various surfaces
and the link with surface properties. In this section, we focus on the particular case of
the wettability of carbonates.
a) Average core-scale wettability
Many studies using the USBM and Amott tests confirm that carbonate reservoir rocks
show intermediate to weakly oil-wet behaviour at the core scale (Cuiec and Yahya, 1991,
Lichaa et al., 1993, Al-Yousef et al., 1995, Skauge et al., 2007, Okasha et al., 2007).
Nonetheless, Okasha et al. (2007) observed an intermediate to slightly water-wet state
for rocks below the OWC. This is consistent with the general trend of higher water-
wetness with decreasing height in the oil column. Typically, while carbonate reservoir
rocks could not be changed further than weakly water-wet state after cleaning, no
matter how strong the cleaning procedure was, their βpreservedβ wettability state could
be readily restored after ageing in crude oil (Cuiec and Yahya, 1991, Lichaa et al., 1993).
Morrow and Buckley (2006) succeeded for the first time to identify a set of carbonate
rocks that exhibited a very strongly water-wet (VSWW) condition after thorough
cleaning. This result is important as it provides a clear base case to analyse the effect of
wettability change on oil recovery.
b) Factors affecting wettability
In fact, the weak affinity of carbonate surfaces to oil is thought to be primarily a result
of acid/base interactions between the slightly acidic components of the crude oil
29
(negatively charged) and the slightly basic carbonate surface (positively charged) (Lichaa
et al., 1993, Al-Yousef et al., 1995).
Morrow and Buckley (2006) thoroughly studied the wettability state of carbonates by
directly measuring contact angles on smooth calcite surfaces using different surface
preparation techniques, crude oils and brines. Overall, they observed that the major
parameter affecting wettability of calcite minerals is the nature of the crude oil. This is
in contrast with an earlier study (Cuiec and Yahya, 1991) which came to the conclusion
that neither the crude oil nor the ageing temperature and pressure affected the
carbonatesβ wettability, but that it was mainly controlled by the surface properties.
Morrow and Buckley (2006) additionally observed that the amount of asphaltenes
exhibited a negative correlation to the measured contact angles, which seems
counterintuitive. They explain this trend by the fact that the kinetics of adsorption are
slow for compounds as heavy as the asphaltenes. In contrast, the effect of the brine
properties was found to be unimportant as no clear trend emerged. In a second series
of experiments, the same authors tested the effect of crude oil composition, initial water
saturation and ageing time on the wettability of two carbonate samples using
spontaneous imbibition curves. They concluded that the acid number of a crude oil is
the most telling property about its wetting alteration potential on carbonate surfaces.
They also confirmed that the ageing time and initial water saturation are major
influential parameters for carbonates, and they exhibit positive and negative
correlations with the wettability alteration, respectively, as expected.
c) Pore-to-pore wettability
Besides, the inter-pore wettability distribution in carbonates is widely recognised as
being non-uniform, where all wettability states (oil-wet, intermediate-wet and water-
wet) may coexist (Cuiec and Yahya, 1991, Lichaa et al., 1993, Al-Yousef et al., 1995). For
instance, Cuiec and Yahya (1991) calculated a fraction of oil-wet surfaces equal to 0.78
and 0.62 for the preserved and restored states, respectively, of a middle-eastern
carbonate rock. However, the wettability distribution within the heterogeneous and
multi-scale pore space of carbonates is largely unknown. There are common claims that
micropores always maintain their strong affinity to water, since they have never been in
contact with oil, and that carbonates exhibit a typical MWL wettability distribution
(Lichaa et al., 1993, Fassi-Fihri et al., 1995). Indeed, Fassi-Fihri et al. (1995) examined a
30
Middle Eastern carbonate reservoir and observed that micropores were water-wet, in
contrast to oil-wet mesopores. Yet oil has been detected within micropores in carbonate
rocks, making oil-wet conditions plausible (Al-Yousef et al., 1995, Clerke, 2009,
Knackstedt et al., 2011, Fung et al., 2011, Clerke et al., 2014, Dodd et al., 2014). Al-
Yousef et al. (1995) performed a centrifugation in crude oil of a cleaned and initially
water-filled microporous carbonate core and observed that most of the microporosity
(diameter ranging between 1 and 10 microns) remained filled with water. Nonetheless,
following a two-weeks ageing and a further centrifugation in crude oil, the latter was
detected in micropores using cryo-SEM imaging.
Skauge et al. (2006) examined the wettability of relatively homogeneous carbonate
samples, each consisting of a unique and different pore class but are mostly
microporous. They found that following core-flooding and a 4-weeks aging procedure,
nearly all the samples exhibited MWS behaviour according to the combined
Amott/USBM test. This means that the smallest pores were oil-wet, which is contrary to
the common assumptions. As an exception, one unique sample was identified as FW
with a weak inclination to MWS.
Marathe et al. (2012) used Field-Emission SEM (FESEM) imaging to identify the existence
of a pattern of wettability alteration within micropores in reservoir carbonates. They
suggested that the wettability alteration on rhombic micrite is face-selective, where the
anhedral (βcurved, rough and poorly formedβ) faces are preferentially oil-wet, as
opposed to the euhedral (βflat, smooth, well-formedβ) which remained water-wet (see
Figure 2.18).
Figure 2.18: FESEM imaging (250 nm scale bars) of the oil deposits (i.e. the footprint of
the wettability alteration) on calcite micro-particles in carbonate rocks (Marathe et al.,
2012).
31
Juri et al. (2016) developed a new approach that characterises the wettability of
carbonate rocks using stochastic inversion of multi-scale PNM to predict relative
permeabilities. They found that not only micropores could be oil-wet but also the
advancing contact angles in micropores could be higher than in larger pores. More
generally, the authors found that the wettability distribution in carbonates was
uncorrelated to pore size.
2.2.3 Effect of wettability on oil recovery
The relationship between wettability and oil recovery for carbonates is poorly
understood. This is mainly due to the heterogeneity of the pore space at multiple scales,
as well as the inability to experimentally restore the reservoirβs initial water saturation
and wettability state. Few studies in the literature attempted to address this issue, in
contrast with a more extensive research on sandstones.
Masalmeh (2002) analysed a broad data from carbonate fields. As expected, they
witnessed the decrease of the residual oil saturation following ageing of water-wet
samples from 20-30% to around 5-10%. Nonetheless, they did not find any clear
correlation of residual oil to initial oil saturations (πππ
πππβ ) for mixed-wet carbonate
samples, as opposed to a linear increase for water-wet sandstones.
Skauge et al. (2006) examined experimentally the oil recovery after waterflooding from
a broad carbonate (mostly microporous) dataset. Interestingly, they observed that
carbonates with low-permeability (and low porosity), which could be related to a high
microporosity content, may exhibit a good oil recovery efficiency (e.g. 45% of the oil was
recovered from a rock with a permeability as low as 0.7mD). This was explained by a
good connectivity of the porous space. Moreover, as expected, the residual oil
saturation was lower at aged conditions (intermediate to slightly oil-wet) than at the
initial cleaned state, for almost all the samples. The change was surprisingly significant
for most of the aged microporous samples, all of these falling into the MWS distribution
(Section 2.1.4), with the recovery factor increasing as much as almost three times the
initial value.
Tie and Morrow (2005) studied the response to wettability change of three initially very
strongly water-wet carbonate samples, two classified as βgrainstonesβ (mud-free and
grain supported) with different heterogeneities and one as βboundstoneβ (bound by its
32
components during deposition). They aged the rocks in crude oil. Then, depending on
the nature of the oil displaced by water during imbibition, called βprobeβ oil, they
obtained three different wettability states:
MXW: typical mixed-wettability state where the crude oil is the βprobeβ oil.
MXW-F: the crude oil is removed first by a solvent, leaving a film of adsorbed
polar components (F); then the solvent is in turn replaced by a mineral oil
(characterised by its low solvency for its asphaltenes) used as βprobeβ.
MXW-DF: the crude oil is directly flooded (DF) by the mineral βprobeβ oil.
Omitting the use of solvent enhances destabilisation and surface precipitation of
asphaltenes by direct contact between crude and mineral oils.
As shown from Figure 2.19, MXW, MXW-F and MXW-DF tend to increase the rockβs oil-
wetness, in sequence, up to neutral-wet conditions. Additionally, they observed that the
two βgrainstonesβ exhibited trends similar to those reported for sandstones by
Jadhunandan and Morrow (1995). In fact, as shown in Figure 2.19, the maximal oil
recovery was reached when wettability was slightly water-wet to intermediate-wet.
Note that following wettability alteration, the oil saturation reached up to half the initial
VSWW value. However, the βboundstoneβ did not exhibit any clear correlation between
wettability and oil recovery. Although this is ascribed to the difference in pore structure,
the variations in the preparation methods should also be taken into account.
Figure 2.19: Oil recovery for three different carbonate rocks (two βgrainstonesβ in black
and blue, and a βboundstoneβ in red) as a function of the Amott-Harvey index. Three
wettability states (MXW, MXW-F and MXW-DF, refer to text) are compared to the
VSWW base case. WW: water-wet; IW: intermediate-wet (WWW: weakly water-wet;
NW: neutral-wet; WOW: weakly oil-wet); OW: oil-wet (Tie and Morrow, 2005).
33
2.2.4 Multi-scale modelling
The pore-space of carbonate rocks is particularly difficult to numerically reproduce due
to the multiple orders of change in the pore-size distribution. Indeed, common imaging
technologies fail to capture the heterogeneous nature of such multi-scale pore systems.
The challenge resides in combining data from distinct scales of porosity obtained
separately using imaging at different resolutions. This issue has been tackled by many
authors in the recent years (Roth et al., 2011, Jiang et al., 2013, Bultreys et al., 2015,
ProdanoviΔ et al., 2015, Harland et al., 2015).
Jiang et al. (2013) suggested a multi-scale pore-network extraction methodology for
reconstruction of multi-scale carbonate rocks. The workflow can be summarised in four
successive steps:
Two 3D pore networks are extracted from micro-CT images at two distinct
resolutions: (i) a fine-scale network extracted at one length-scale and (ii) a
coarse-scale network extracted at a higher length-scale. Note that the two
networks may have some overlap in pore sizes.
A sub-network is extracted from the original coarse-scale network, its domain
corresponding to the nesting domain for the target multi-scale network.
A stochastic fine-scale network is statistically generated within the same nesting
domain, without overlapping with the elements of the coarse-scale network. It
is equivalent to the original network in terms of geometry, topology and
correlations.
The fine-scale and coarse-scale networks are integrated into a single two-scale
network. Indeed, after superimposing the two networks, the fine-scale bonds
intersecting with the coarse-scale bonds are suppressed. Then, the cross-scale
connections are ensured by adding connecting fine-scale bonds between coarse-
scale nodes and neighbouring fine-scale nodes.
Bultreys et al. (2015) developed a new pore network modelling technique that couples
the microporosity and the macroporosity. It consists in incorporating into a standard
macroporous network some notional connecting micropores or βmicro-linksβ. The latter
are dealt with as a continuous porous medium with pre-assigned petrophysical
34
properties, and may join-up initially disconnected macroporous networks to allow
drainage to occur.
Harland et al. (2015) developed an original object-based rock reconstruction
methodology which reproduces the micro-scale structure of microporous carbonates.
The workflow consists in the analysis of 2D SEM image data of carbonate samples to
capture the basic structural features of micropores. This information is then used to
populate 3D digital images with micrite crystals of varying shapes, sizes and packing
patterns. Afterwards, single-phase and multi-phase simulations are run on the obtained
3D image (using Lattice-Boltzmann) and a network extracted from it (using PNM),
respectively. They showed that the percentage of microporosity among the total
porosity strongly influenced the single-phase permeability. For multi-phase imbibition
calculations, they considered either a uniformly water-wet distribution or fractionally-
wet (FW), with half of the pores oil-wet and the other half water-wet. The authors
showed that both the wettability distribution and the networkβs homogeneity i.e.
proportion of microporosity, affected the oil recovery. Indeed, increasing the networkβs
homogeneity in water-wet conditions resulted in a better recovery due to a higher
sweep efficiency, which is in accordance with earlier observations (Clerke, 2009, Clerke
et al., 2014).
Note that micropores may supply the connectivity and fluid pathway between otherwise
disconnected larger pores within some multi-scale pore network models (Roth et al.,
2011, Jiang et al., 2013, Bultreys et al., 2015).
2.3 Discussion
Having reviewed the literature on wettability and microporous carbonates, we identify
some needs that few studies have attempted to address, which coincide with our
objectives stated in Chapter 1:
a) Model the wettability distribution patterns identified by Marathe et al. (2012) in
microporous carbonates, where the wettability is dependent on the pore shape.
b) Assess the impact of the wettability of micropores on oil recovery in multi-scale
carbonate networks, especially when micropores control the networkβs
connectivity.
35
c) Suggest a pore-scale mechanism that may be responsible for oil migration into
initially strongly-water-wet micropores. Actually, an oil-wet state in micropores
may have developed over geological time either (i) by means of calcite
cementation as some pores get smaller, or (ii) in the upper regions of
exceptionally large oil columns where high enough capillary pressures are
reached, or (iii) if the micropores undergo progressive wettability changes during
oil migration, which reduces their capillary entry level and makes them more
prone to oil invasion. Scenario (i) is the least likely to have happened, since as
previously stated in Section 2.2.1, the cementation process is slowed down in the
presence of the oleic phase. We will focus in Chapter 5 on scenario (iii), which we
think is perfectly plausible.
36
Chapter 3 : Pore network model
3.1 Introduction
Pore Network Modelling (PNM) is a physically-based multi-phase flow modelling tool
that offers a computationally-cheap simulation workflow to derive petrophysical
properties e.g. relative permeabilities, capillary pressures, residual oil saturations, etc.
Their input networks are approximations of the pore space, either generated
numerically or extracted from 3D images of real core samples (see Figure 3.1). The utility
of Pore Network Models resides in the ability to incorporate in their simplified
geometries appropriate pore-level physics of displacement mechanisms, e.g. fluid films
and layers, etc. This leads to a better understanding of complex flow and transport
mechanisms by allowing to βexamine the expected consequences of various pore scale
phenomena at the macroscopic scaleβ (Dixit et al., 1999). Indeed, PNM assists reservoir
engineers in gaining insight into laboratory experiments, reservoir behaviour and
recovery mechanisms, by deriving and explaining macroscopic properties. Additionally,
compared to Direct Numerical Simulations, it is computationally cheaper, meaning that
it can support bigger models (larger volume and range of pore sizes), thus providing a
better βrepresentativityβ of the core sample.
Figure 3.1: Different workflows leading from a core sample to relative permeability and
capillary pressure curves, after Ryazanov (2012).
37
3.2 Pore Network modelling tool
The quasi-static two-phase flow pore network modelling tool used in this study is similar
to previously developed models (Oren et al., 1998, Patzek, 2001, Valvatne and Blunt,
2004). It has been thoroughly described by Ryazanov (Ryazanov et al., 2009, Ryazanov,
2012, Ryazanov et al., 2014). It requires as input networks consisting of pore bodies
(nodes) connected by pore throats (bonds). Throughout this work, we generically refer
to both pore bodies and pore throats as pores. The porosity of the network is simply
estimated using the total volume of pores relative to the volume of the enclosing
domain. Permeabilities are computed based on the total flow through the network for
the phase cluster of interest, for a given applied pressure gradient and application of
Darcyβs law.
The input networks can be either synthetically generated or extracted from 3D micro-
CT images. Extracted networks are topologically and geometrically equivalent
representations of the porous media. On the one hand, they preserve the pore-space
connectivity through identification of the nodes that represent the main junctions; and
the bonds that consist of the remaining narrower connections. On the other hand,
network extraction preserves the main geometrical pore properties (inscribed and
hydraulic radii, shape factor, volume, etc.) by idealising pore shapes into straight tubes
with simple cross-sections (e.g. circle, triangle, star). This leads to easy analytical
calculations of flow properties.
3.2.1 Pore shapes
The shape of the pore cross-section is important for conductance and capillary entry
pressure computations. Additionally, pore shapes are central to the models that we
developed in the following chapters, since they may control the wettability and primary
drainage patterns. Although in earlier works, pore shapes have been approximated as
circles, these fail to capture important features such as corner films and layers found in
angular pores. Our pore network simulator takes into account a wide range of pore sizes,
including circles, triangles, squares and n-cornered stars. This section summarises the
methodology for idealising the pore shapes using information from the input network.
The equations for single and two-phase conductances and the capillary entry pressures
for each of the idealised shapes were thoroughly described by Ryazanov (2012).
38
a) Circle-Triangle-Square (CTS)
The pores cross-sectional shape is usually characterised using the shape factor, πΊ, to
idealise it as a Circle, Triangle or Square (the CTS approach). The shape factor is defined
as πΊ = π΄πΏ2β , where π΄ and L are the pore cross-section area and perimeter, respectively.
Depending on the value of πΊ, the cross-sectional shape is approximated as one of the
idealised shapes, using the following rules:
if πΊ β€β3
36: (Arbitrary) Triangle, keeping the same G.
if β3
36β€ πΊ β€
1
16: Square with πΊ =
1
16 (approximation).
if 1
16< πΊ: Circle with πΊ =
1
4π (approximation).
The main shortcoming of the CTS method is that it leads strictly to convex pores, thus
fails to represent non-convex pores that exist in real pore spaces. We may use the
dimensionless hydraulic radius, π», as an indicator of the convexity of a shape (Ryazanov,
2012). π» is defined as π» = (π΄/πΏ)/π πππ , where π΄ and πΏ are as defined above, and π πππ
is the inscribed radius. Notice that shapes with π» β₯ 1/2 are (generally) convex. Since all
the CTS shapes are characterised by π» = 1/2, they are labelled as convex.
b) n-cornered star shapes
In addition to the shape factor, we use the previously defined dimensionless hydraulic
radius, π», to idealise the pore shape as regular non-convex n-cornered stars, along with
the CTS (Helland et al., 2008, Ryazanov et al., 2009, Ryazanov, 2012, Ryazanov et al.,
2014). As shown in Figure 3.2, regular n-cornered stars are characterised by two
parameters, the number of corners, n, and the half-angle Ξ³. While several (n, Ξ³) pairs
may exist for a given shape factor πΊ, a given (πΊ, π») pair generally satisfies a unique (n, Ξ³)
i.e. a single regular star shape (Helland et al., 2008). However, the working network
needs to specifically include an extra information on the hydraulic radius. Practically,
this corresponds to an additional column in the input file.
39
Figure 3.2: An example of a n-cornered star shape (π = 5), with inscribed and hydraulic
radii π πππ and π β, respectively, and half-angle πΎ.
The methodology for shapes characterisation from the (πΊ, π») pair is illustrated in Figure
3.3, and described as follows:
For π» < 1/2, the curves correspond to star shapes with different number of
corners. Keeping the same value of the shape factor πΊ as the initial arbitrary
shape, two number of corners values, π1 and π2 are possible. The nearest n-
cornered star curve is chosen, which corresponds to an approximated π», slightly
different from the initial shapeβs value. The half-angle πΎ is then deduced, and the
idealised shape is fully characterised.
The vertical line π» = 1/2 represents the arbitrary triangles (π = 3), along with
regular shapes like n-cornered polygons, including squares (π = 4) and circles
(π = +β). Note that the polygons are a limiting case of stars, satisfying the
following condition: πΎ =π
2β
π
π.
The curves at π» > 1/2 correspond to some more convex shapes that will not be
taken into account in this study.
40
Figure 3.3: Shape factor, πΊ, as a function of dimensionless hydraulic radius, π», for some
cross-sectional shapes. The right and left boundaries represent the shapes theoretical
limiting (πΊ, π») pairs (Ryazanov, 2012).
3.2.2 Fluid configuration
The network modelling tool is a two-phase flow model, for which oil and water are the
two phases of interest. The fluid configuration of every pore may consist of different
fluid elements, such as oil/water bulk (centre of the pore), oil/water corner films and oil
layers (sandwiched between water in the corners and bulk). All the possible cross-
sectional fluid configurations incorporated in the model are shown in Figure 3.4. Note
that we define a phase cluster as an interconnected set of pores that contains at least
one fluid element of the corresponding phase. Since a single pore may contain elements
from each of the two phases, it can be part of both an oil and a water cluster.
41
Figure 3.4: Possible cross-sectional fluid configurations following primary drainage and
imbibition for regular n-cornered star shapes (represented by equilateral triangles for
simplicity). Oil is red; water is blue and surfaces of altered wettability are brown
(Ryazanov, 2012).
3.2.3 Pore-level displacements
The displacement process is quasi-static, as the capillary forces are assumed to
dominate over viscous forces i.e. the capillary number satisfies the following condition
(Blunt and Scher, 1995):
ππ =ππ
πππ€β€ 10β4 (3.1)
where π is the Darcy flux, π is the viscosity and πππ€ the oil-water interfacial tension. The
capillary-driven flow is simulated using a classical Invasion Percolation (IP) process with
trapping, where an invading phase is eligible to displace a defending phase from a pore
if the pore satisfies two conditions:
i. accessible, i.e. it has at least a neighbouring pore that is part of the invading
cluster and is connected to the inlet; we may distinguish between a strong or a
weak accessibility condition, where the connection to the inlet is exclusively
42
fulfilled through a neighbouring bulk fluid element or can additionally be realised
through a neighbouring film, respectively.
ii. non-trapped, i.e. it is part of a defending cluster that is connected to the outlet.
The quasi-static IP process is driven by a gradual change of the overall capillary pressure.
Each capillary pressure corresponds to an equilibrium fluid configuration within a pore
(Figure 3.4). This is determined by pore-level displacements that occur sequentially
according to well-defined capillary pressure criteria. Indeed, the so-called MS-P theory
(Mayer and Stowe 1965) assigns to each pore displacement a capillary entry
pressure, ππππ‘ππ¦ =πππ€
π πππ‘ππ¦, where π πππ‘ππ¦ is the entry radius of the specific shape and
displacement. A pore displacement occurs when its entry pressure is overcome and the
pore satisfies conditions i. and ii. described above.
Lenormand et al. (1983) identified 3 distinct displacement mechanisms as shown in
Figure 3.5: Piston-like (PL), snap-off (SO) and pore body filling (PBF). Further, we identify
a 4th distinct pore-displacement, namely the oil layers formation and collapse
mechanism. Note that when the invading phase is the non-wetting wetting phase, it is
called drainage. Otherwise, we refer to it as imbibition. Further, for the sake of
simplicity, we consider water to be the wetting phase and oil the non-wetting phase in
the description below.
Figure 3.5: Pore-level displacement mechanisms, (a) piston-like, (b) snap-off, (c) and (d)
I1 and I2 PBF events, with 1 and 2 adjacent pores filled with the non-wetting oil phase,
respectively, as observed from micromodel experiments by Lenormand et al. (1983).
43
a) Piston-like (PL)
In this type of displacement, the bulk fluid element is displaced in a piston-fashion from
the centre of the pore. It is the only displacement modelled during drainage, and can
occur as well during imbibition. PL requires a strong accessibility condition i.e. the pore
needs to have at least a neighbouring pore that is bulk-filled with the invading phase
and connected to the inlet; Note that a phenomenon, that we refer to as βbypassingβ, is
likely to occur during imbibition when water in the corner is present. As illustrated in
Figure 3.6(a), it consists of a PL filling when the neighbouring pore that is bulk-filled with
water is connected to the inlet through corner water rather than bulk. βBypassingβ tends
to create trapped oil clusters, hence to reduce the residual oil saturation, in contrast to
an efficient oil sweep from the inlet when water in the corners is absent (Figure 3.6(b)).
Figure 3.6: Illustration of different water invasion patterns, depending whether water
in the corners is (a) present or (b) absent. In the first case, βbypassingβ is likely to occur,
which tends to create trapped oil clusters, as opposed to an efficient oil sweep from the
inlet for the second case. Note that water (blue) displaces oil (red) from the inlet (left)
to the outlet (right) of a 2D regular network.
44
b) Snap-off (SO)
Snap-off is an imbibition process that consists of the swelling of the water film until
destabilisation of the oil/water interface and eventual water-invasion of the pore (Figure
3.5(b)). Unlike PL, SO requires a weak accessibility condition, since it may take place
within a pore where water is only connected through films (rather than bulk) to the inlet.
Nonetheless, SO is less favoured than PL in terms of entry pressures. Consequently, SO
will only occur if there is no adjacent element whose centre is filled with water.
Additionally, SO usually occurs under strongly water-wet conditions, is less likely to
happen at weakly water-wet states, and is inhibited at intermediate and oil-wet
conditions.
SO is considered to be the major mechanism for trapping of the non-wetting phase in
the imbibition process since it tends to create disconnected non-wetting clusters all
through the network. This leads to high levels of residual oil saturation.
c) Pore Body Filling (PBF)
Pore body filling (PBF), as illustrated in Figure 3.5(c)-(d), is an imbibition mechanism that
corresponds to the displacement of the non-wetting oil phase from the centre of a node
by the cooperative interfaces movement from the adjacent water-filled bonds.
If the node is directly connected to π§ bonds, π§ β 1 possible entry pressures related to
πΌπ (π β [1, π§ β 1]) events are computed beforehand, different from each other, using a
bunch of parametric models available in the literature (Blunt, 1997, Blunt, 1998, Oren
et al., 1998, Patzek, 2001, Valvatne and Blunt, 2004). Each PBF event πΌπ corresponds to
a distinct possible configuration related to the number of connected oil-filled bonds π
at the time of invasion. Notice that the displacement πΌ1 coincides with the PL
displacement, and is the most favoured event (highest ππππ‘ππ¦). Moreover, the πΌπ entry
pressures should be positive, hence the PBF only occurs for spontaneous imbibition
(ππ β₯ 0).
Note that multiple PBF models have been previously implemented within the pore
network model (Ryazanov, 2012). However, none of these models is rigorous, but rather
based on arbitrary assumptions. In fact, they depend on random parameters and ad-hoc
weighing coefficients. Hence, we decided to exclude the PBF events from this work.
45
d) Oil layers formation and collapse
During imbibition, oil layers may form in pores where parts of the surface are altered to
strongly oil-wet; they are sandwiched between water in the centre and corners of the
pores (Figure 3.7). While the contribution of these layers to oil volume and flow is small
compared to that of bulk fluid, they play an important role in preserving the oil phase
connectivity and driving it to very low saturations. However, they may get destabilised
and collapse at a critical capillary pressure. In previous pore network (Blunt, 1998,
Patzek, 2001, Valvatne and Blunt, 2004), the collapse of oil layers depends on a
geometrical criterion as it takes place when the surrounding oil/water interfaces meet.
In our simulator, the formation and collapse of the layers is more physically consistent,
as it occurs according to a realistic thermodynamic criterion (van Dijke and Sorbie, 2006).
Figure 3.7: Illustration of the oil-layer configuration at the corner of a triangular cross-
section, bounded by inner and outer arc menisci AM1 and AM2, with contact angles
πβand ππ, respectively. πΏπ ππ is the distance of AM1 from the corner edge. πΎ is the half-
angle. Oil is red, water is blue and surfaces of altered wettability are brown.
3.2.4 Flooding cycle
We simulate the commonly used flooding cycle: primary drainage, ageing and water
invasion, which mimics the flooding process undergone by a real hydrocarbon reservoir.
a) Primary Drainage (PD)
First, PD is simulated in a network where all pores are initially fully saturated with water
and water-wet. In a first typical approach in Chapter 4, each pore is assigned a single
46
water-wet contact angle πππ that ranges between 0 and 90Β° and is chosen to be the
same throughout the network. Alternatively, we developed in Chapter 5 a novel
approach where each poreβs contact angle evolves during primary drainage, depending
on the concentration of polar compounds in the oil phase.
To establish the primary drainage pore-filling sequence, pore displacements entry
pressures, ππππ‘ππ¦, are explicitly computed for each pore prior to the PD simulation. The
equations are detailed by Ryazanov (2012). Only one of two possible displacements is
computed for each n-cornered pore, depending on its shape (πΎ) and wettability (πππ):
If πππ <π
2β πΎ, the pore is sufficiently water-wet to retain water in the corners,
hence a PL displacement with formation of water corner films applies, called A-
Dr (corresponds to the change in fluid configuration a->c in Figure 3.4).
If πππ β₯π
2β πΎ, a PL displacement with the complete removal of water from the
pore, B-Dr, applies (a->b in Figure 3.4).
To decide the preliminary sequence of invasion events, the displacements for all the
pores are then sorted in order of increasing ππππ‘ππ¦ .
During PD, while the inlet oil phase pressure, ππ , is increased in the inlet, the outlet
water phase pressure, ππ€ is kept constant. This results in the gradual increase of the
overall network capillary pressure, ππ = ππ β ππ€. During this process, accessible and
non-trapped bulk water-filled pores with the lowest ππππ‘ππ¦ (which turn below ππ) are
invaded first by oil according to either A-Dr or B-Dr. Only 3 fluid configurations are
possible for each pore (π, π and π in Figure 3.4). Note that as long as ππ is increasing, the
corner water films with an outlet connection keep shrinking, as the length of contact of
the water film with the solid in a corner, πΏπ , decreases in accordance with Equation (3.2).
This results in a continuous decrease in the water saturation.
πΏπ =πππ€
ππβ
cos (πππ + πΎ)
sin πΎ (3.2)
The process of oil invasion may be artificially stopped at a fixed initial water saturation,
ππ€π, related to a predefined maximum capillary pressure, πππππ₯. We refer to it as ππ€π at
πππππ₯. Alternatively, PD may continue until the irreducible water saturation is reached
47
where no further pore-scale displacements are possible under capillary controlled
processes. We refer to it as ππ€π at πππππ₯ β +β. Note that in the latter case, ππ
πππ₯ is
practically finite, corresponding to the highest entry pressure amongst the invaded
pores. Additionally, notice that while the corner water films generated by displacement
A-Dr maintain the water phase connectivity and may lead to water saturations close to
zero, their absence following B-Dr displacement generates trapping of water clusters.
Hence, the water structure at the end of PD consists of corner films and trapped bulk
elements.
b) Wettability Alteration (ageing)
Subsequent to PD, the model allows for wettability alteration (ageing) of the oil-filled
pores. This has consequences at two different scales i.e. intra-pore and inter-pore, as
described in Figure 2.9, and further explained below.
Inter-pore scale
Ageing may result in a mixed-wettability state within the network. In fact, each pore is
assigned either a water-wet advancing contact angle, ππ,π€π€π[πππ , 90Β°[, or an oil-wet
advancing contact angle, ππ,ππ€π]90Β°, 180Β°]. This allocation depends on two predefined
parameters: the oil-wet fraction, πππ€, and the distribution type. The former is defined as
the volumetric fraction of oil-wet pores among all the pores in the network, theoretically
ranging between 0 and 1. For the latter, we choose from the commonly used wettability
alteration distributions (Section 2.1.4), the pore-size related mixed-wet large (MWL) and
mixed-wet small (MWS) distributions, or to the fractionally-wet (FW) size-independent
distribution. Alternatively, we have implemented a physically based wettability
alteration scenario that depends on both pore size and shape Chapter 4, leading to the
Altered-Wet (AW) distribution.
Intra-pore scale
Ageing may result in a non-uniform wettability within a single oil-filled pore (Kovscek et
al., 1993). Indeed, due to polar species adsorption from the oil, only the wettability of
oil-contacted surfaces changes. These surfaces acquire an advancing (either water-wet
or oil-wet) contact angle ππ β₯ πππ. Depending on the initial fluid configuration within
the pore following PD, two transitions are possible: π β π or π β π (Figure 3.4). In the
48
latter case, different parts of the pore wall have different contact angles, with ππ
assigned to the central surfaces and πππ to the corners area.
c) Waterflooding
Pore displacements
Pore displacement entry pressures, ππππ‘ππ¦, are computed for each pore before
waterflooding, some explicitly and others iteratively, using equations described by
Ryazanov (2012). Different displacement types exist for each n-cornered star-shaped
pore, depending on its shapeβs corner half angle (πΎ, refer to Figure 3.2), wettability (ππ
and πππ) and accessibility.
Initial configuration with corner water
If πππ <π
2β πΎ, water films exist in the corners prior to waterflooding (configuration d in
Figure 3.4). For this case, distinct PL displacements may be possible
o If ππ β₯π
2β πΎ: two successive displacements are considered:
PL with the formation of oil layer, sandwiched between water
in the corner and bulk, A-Imb (π β π in Figure 3.4).
Oil layers collapse, according to a thermodynamic criterion, C-
Imb (π β π in Figure 3.4).
Note that A-Imb and C-imb only exist together, and a condition for this is
that ππππ‘ππ¦π΄βπΌππ > ππππ‘ππ¦
πΆβπΌππ. The range of contact angles and capillary
pressures for which layers exist has been thoroughly examined by
Ryazanov (2012). If the condition is not satisfied, the displacement B-Imb
below applies.
o If ππ <π
2β πΎ or ππππ‘ππ¦
π΄βπΌππ > ππππ‘ππ¦πΆβπΌππ: PL with complete removal of oil
applies, B-Imb (π β π in Figure 3.4).
Note that either of the PL displacements A-Imb (and C-Imb) or B-Imb are stored in the
poreβs list of possible displacements. Additionally, regardless of the value of ππ, an extra
snap-off displacement is computed for each pore and added to the list. Depending on
the pore accessibility during waterflooding, either SO or PL will take place.
49
o Snap-off, D-Imb (π β π in Figure 3.4). As stated above (Section b)),
ππππ‘ππ¦ππ < ππππ‘ππ¦
ππΏ ;
Initial configuration without corner water
If πππ β₯π
2β πΎ, the pore is initially fully saturated with oil (configuration j in Figure 3.4).
Similar to PD (but with swapped phases), two displacements are possible, depending on
πΎ and ππ:
o If ππ >π
2β πΎ: the pore is sufficiently oil-wet to retain oil in the corners,
thus a PL displacement with formation of oil corner films PL applies,
called A1-Imb (π β π in Figure 3.4).
o If ππ β€π
2β πΎ: a PL displacement that completely removes oil from the
pore, B2-Imb, is computed (π β π in Figure 3.4).
Eventually, the preliminary invasion sequence prior to water invasion for all the pores is
sorted in order of decreasing displacement entry pressures
Invasion process
At the start of waterflooding, the length of the contact of water with the solid in the
corner is fixed at πΏπ ππ, computed from Equation (3.2) at ππ = ππ
πππ₯ . Following ageing,
the jump in contact angles (πππ β ππ) at πΏπ ππ leads to the pinning of the meniscus AM1
(Morrow, 1990, Kovscek et al., 1993) with a related βhingingβ contact angle, πβ, as
illustrated in Figure 3.7. While the overall network ππ is decreased, πβ increases from
πππ to ππ, corresponding to the change in fluid configuration π β π in Figure 3.4. As the
waterflood carries on, accessible and non-trapped pores from the ordered
displacements list whose Pπππ‘ππ¦ are above ππ are invaded by water, in accordance to
either A, A1, B, B1, C or D-Imb. Note that in the particular case of ππ being low enough
(ππ <π
2β πΎ), πβ may overcome ππ, then AM1 may start moving towards the surface
with altered wettability at the centre of the pore (π β β in Figure 3.4). Particularly, if
the AM1s meet at the centre (configuration π in Figure 3.4), they are destabilised and a
particular case of snap-off occurs (π β π in Figure 3.4).
50
In our simulations, we assume that the process is carried on until the residual oil
saturation, πππ, is reached, below which no further oil mobilisation can take place by
capillary forces (πππ at πππππ β ββ).
3.2.5 Network and phases connectivity
The Euler number π is a topological invariant utilised as a measure of the connectivity
of a fluid phase (either oil or water) within a network (Vogel, 2002, Herring et al., 2013).
The fluid phase may consist of both bulk and film elements. The Euler number can be
calculated as π = π½0 β π½1, where π½0 is the 0th Betti number, or the number of separate
fluid phase elements in the network; and π½1 is the 1st Betti number, consisting of the
number of redundant connections or loops in the fluid phase structure. π½1 can be
regarded as the maximum number of cuts that can be applied to the fluid phase
structure without breaking it into separate parts. Note that by definition, the more
negative is π, the more connected is the structure.
The Euler number of the network as a whole, considered as if fully saturated with only
one phase, is simply computed as ππππ‘ = ππ β ππ΅ , where ππ is the number of nodes,
including the βnotionalβ boundary nodes, and ππ΅ is the number of bonds. As for the
Euler number of a particular phase of interest βπββ, it is calculated as ππβ = πππββ
ππ΅πβ, where ππ΅πβ
is the number of bonds filled with phase "πβ"; and πππβ is the
number of nodes, either βπββ-filled, or directly adjacent to a βπββ-filled bond (including
the βnotionalβ boundary nodes). Note that in the particular case where two βπββ-filled
bonds share a node that is filled with the other phase, the node is counted twice. An
example of Euler number calculations is performed in Figure 3.8.
51
Figure 3.8: An example of Euler number calculations carried out on an illustrative 2D
regular cubic network consisting of spherical nodes and tubular bonds. Periodic
boundary conditions apply between the bottom and top. Oil (red) displaced water
(blue) from the inlet (left) to the outlet (right); the βnotionalβ boundary nodes are
illustrated as dashed hollow circles. The network is a single object (π½0 =1) with 7
redundant loops (π½1 = 7). Further, ππ = 15 (9 actual + 6 boundary nodes) and ππ΅ =
21; hence ππππ‘ = π½0 β π½1 = ππ β ππ΅ = β6. Similarly, for the oil phase,
(π½0πππ, π½1πππ
) = (3, 2) and (πππππ, ππ΅πππ
) = (14, 13), hence ππππ = 1.
The Euler number is used in Section 3.3 below to derive the connectivity function of a
given network, which describes its connectivity as a function of the pores size (Vogel,
2002). Besides, in the next chapters, we use the normalized Euler number οΏ½ΜοΏ½ as a
measure of the connectivity of a fluid phase within the network (Herring et al., 2013).
For a given phase (either oil or water), ππβΜ =ππβ
ππππ‘ is the ratio of the Euler number of the
selected phase over the networkβs Euler number. By definition, ππβΜ = 1 corresponds to
maximum phase connectivity, while ππβΜ β€ 0 corresponds to a disconnected phase
(ππβ β₯ 0) within a connected network (ππππ‘ < 0).
3.3 Input networks
In this thesis, two input networks with different topologies will be used: a homogeneous
Berea sandstone network with a relatively narrow range of pore sizes, and a
heterogeneous two-scale carbonate network with a wide range of pore sizes, where the
smallest pores provide overall connectivity to the otherwise disconnected network.
52
3.3.1 Berea sandstone network
The network (Figure 3.9(a)) has been extracted from a 3D micro-computed tomography
image of a Berea sandstone sample, using the enhanced extraction technique described
by Jiang et al. (2007). Its characteristics are summarized in Table 3.1. Figure 3.9(b) shows
that the pore size distribution is unimodal, and that larger pores tend to have more
corners.
(a) (b)
Figure 3.9: Berea network: (a) 3D representation and (b) pore size (inscribed radius)
and shape distributions.
Number of pore elements (nodes and bonds) 22,251
Average coordination number 3.7
Porosity (%): 18.97
Absolute Permeability (mD): 1576.38
Table 3.1: Main properties of the Berea network.
3.3.2 Multiscale carbonate network
We present here a heterogeneous network (Figure 3.10) constructed from a multiscale
dataset for a microporous carbonate, and whose characteristics are summarised in
Table 3.2. The methodology and workflow of the two-scale network generation have
been presented by Jiang et al. (2013) and summarised in Section 2.2.4. Note that the
coarse scale network clearly lacks overall connectivity (Figure 3.10(b)), and has only
become connected through integration with the well-connected fine scale network
(Figure 3.10(a)).
53
(a) (b) (c)
Figure 3.10: (a) Statistically generated fine network extracted from a micro-CT image at
2.86 ππ resolution, (b) coarse network extracted from another micro-CT image at
14.29 ππ resolution and (c) the resulting integrated two-scale network. Note that both
images are derived from the same microporous carbonate dataset.
Number of pore elements 26,349
Average coordination number 3.5
Porosity (%): 21
Absolute Permeability (mD): 59
Table 3.2: Main properties of the carbonate network.
Figure 3.11 indicates a wide distribution of pore radii in the resulting two-scale network,
ranging from 1 to 146 Β΅m, and that micropores (π πππ < 5ππ) represent around 15 % of
the network volume (corresponding to the first bin in the pore size distribution).
Additionally, note that the micropores are represented by a wide range of cross-
sectional shapes.
The connectivity functions of the carbonate and Berea networks are compared in Figure
3.12. Each one is obtained by removing pores from the network in order of increasing
size and re-computing ππππ‘ at each step. Suppressing larger and larger pores leads to a
decrease in connectivity i.e. increase in ππππ‘ from negative values. The radius at which
the network (just) gets disconnected i.e. ππππ‘ = 0, may correspond to the networkβs
percolation radius. Figure 3.12 shows that the carbonate networkβs percolation radius
of around 5 ππ is much smaller than the Bereaβs value of 13 ππ. This further confirms
that the carbonate network connectivity is mainly controlled by its micropores.
54
Figure 3.11: Carbonate network pore size (inscribed radius) and shape distributions.
Figure 3.12: Comparison between the connectivity functions of the carbonate and
Berea networks, where ππππ‘/ππππ‘ is the Euler number of the network divided by its
total volume.
3.4 Conclusion
In this chapter, we described the pore network multi-phase model that we use
throughout this work. The model has two main specifics compared to other pore
network models available from the literature:
Involves a wider variety of cross-sectional pore shapes, mainly the N-star shapes,
as compared to standard CTS (Circle-Triangle-Square) shapes. This leads to a
broad range of possible fluid configurations and the corresponding pore-scale
displacements;
Incorporates a physically-based thermodynamic criterion for oil layer formation
and collapse, instead of the previously used geometrical criterion.
Note that the model described in this chapter was created by Ryazanov (2012), except
for the new connectivity calculations using the Euler number. In the following chapters,
55
we describe the two novel models that we have incorporated in the original pore
network model as part of this thesis:
Scenario 1: Wettability alteration following primary drainage and subsequent to
ageing (Chapter 4). This scenario is faithful to the traditional flooding cycle
described in Section 3.2.4. We incorporated into it a physically-based wettability
distribution, based on Kovscek et al. (1993)βs model, that takes into account both
the pore size and shape.
Scenario 2: Wettability alteration starting during primary drainage (Chapter 5).
In this scenario, we modify the way primary drainage is modelled. Indeed, based
on the work by Bennett et al. (2004), we model a wettability alteration that
occurs as the primary drainage occurs, as a result of the adsorption of polar
species from the oil. This wettability evolution changes the primary drainage
patterns.
56
Chapter 4 : Scenario 1 β Wettability alteration following ageing
4.1 Introduction
In this chapter, we implemented the Kovscek et al. (1993)βs model in our pore network
model that takes into account a variety of pore shapes, representative of the complexity
of real carbonate rocks. In Section 4.2, we present the model where equivalent pore wall
curvatures are assigned to pores based on their size and shape. Moreover, the disjoining
pressure is inferred from the fraction of oil-wet pores and the maximum achieved
capillary pressure. In Section 4.3 we investigate the role of the wettability distribution
in oil recovery for pore space structures of different complexities, starting with the
relatively simple Berea sandstone network, which is then compared with the complex
multiscale carbonate network. For the carbonate case, we further examine the
importance of the wettability state of the micropores for oil recovery.
4.2 Model description
We model the physically-based wettability alteration scenario suggested by Kovscek et
al. (1993), thoroughly described in Section 2.1.1, for which the wettability is controlled
by the stability of the thin films through the disjoining pressure. All pores are initially
assumed to be filled with water and completely water-wet, and at this point we start
injecting oil. At the end of the primary drainage, a maximum capillary pressure, πππππ₯,
is reached, either predefined, or corresponding to the highest entry pressure amongst
the oil-invaded pores. For each of the pores invaded, if the oil phase is still connected
to the outlet i.e. not-trapped, the local capillary pressure reached coincides with the
networkβs πππππ₯. Otherwise, it corresponds to the capillary pressure at which trapping
occurred. Besides, the local threshold capillary pressure, ππβ, is computed for each
invaded pore using Equation (2.2). If ππβ was overcome by the local maximum capillary
pressure, we assume that the thin film had collapsed and the pore is rendered oil-wet
following ageing. Otherwise, the pore remains water-wet.
However, in our pore network model, pore walls are represented by flat surfaces with
zero curvature (regular polygon and star shapes), hence thin films would collapse
simultaneously in all pores, as all ππβ are equal for constant Πππππ‘ (Equation (2.2)). As it
is difficult to obtain the real pore wall curvatures from a digitised image, we assign
equivalent curvatures to the flat pore walls based on the overall pore shapes, for
57
wettability alteration purposes only, as indicated in Figure 4.1. The curvature, π = β1
ππ,
is assumed negative as the radius of curvature, ππ, is located outside the shapes. For
regular polygons, ππ is computed using Equation (4.1) (Joekar-Niasar et al., 2010) in
which cosπ is randomly chosen. For regular stars, ππ is computed from Equation (4.2).
Note that while one random parameter (cosπ) is introduced for polygons, the radius of
curvature is fully determined for stars.
ππ
πβππππ¦πππ= π πππ
sinππ
cosπ β sinππ
(4.1)
πππβπ π‘ππ = π β²πππ
sinππ
cosπΎ β sinππ
= π πππ
sin (πΎ +ππ) π ππ
ππ
sin (πΎ +ππ) . cosπΎ β sin
ππ
(4.2)
where π πππ and π πππ β² denote the original and new inscribed radii, respectively;
πππ π ranges between π πππ
π and 1; and πΎ is the corner half-angle of the pore.
(a) (b)
Figure 4.1: Equivalent pore wall curvature assignment for (a) n-cornered Polygon and
(b) n-cornered Star shape, where π πππ and π πππ β² denote the original and new inscribed
radii, respectively; ππ denotes the radius of curvature and π the angle between the
tangent to the newly obtained (red) curved shape at a vertex and the line connecting
the vertex to the centre (π coincides with the corner half-angle, πΎ, for the original
shapes).
According to Equation (2.2), by strictly using the negatively-curved star shapes
introduced in Figure 4.1, the more curved the pore surface is (in absolute value), the
58
more accessible (lower) is the capillary pressure at which the thin film collapses (ππβ).
Therefore, smaller pores of the same shape are more likely to be oil-wet. This is also
the case for pores of the same size and shape but with a larger number of corners,
which can be interpreted as corresponding to increased pore roughness. Therefore, the
model qualitatively reproduces the pattern shown by high-resolution imaging in
carbonate rocks (Marathe et al., 2012) where oil deposition on calcite microparticles
was limited to the anhedral (curved, rough and poorly formed) faces.
Πππππ‘ depends on the fluid system and mineralogy of the rock surface. We will assume
in our model that it is constant throughout the network, reflecting a mono-mineral rock.
Πππππ‘ is computed by the simulator, given a prescribed final drainage capillary pressure,
πππππ₯ , and a targeted volumetric oil-wet fraction, πππ€, the ratio of the oil-wet to the
total pore volumes. Note that a higher πππππ₯ leads to a higher πππ€, as thinner films are
more prone to collapse. On the other hand, higher Πππππ‘ causes πππ€ to decrease, since
a higher ππ would need to be achieved to reach film rupture (at ππβ ).
Throughout the thesis, we refer to the resulting wettability distribution as Altered-Wet
(AW). Below (Sections 4.3.1(b) and 4.3.2(b)) we describe how this new AW distribution
compares to previously described wettability distributions MWL, MWS and FW.
Note that since no comprehensive model is available for the distribution of the values
of the advancing contact angle ππ, these are assumed to be uniformly distributed within
prescribed ranges in both the water-wet and oil-wet pores.
4.3 Results and discussion
The simulations are carried out on the two distinctly different networks presented in
Section 3.3: a homogeneous sandstone network and a heterogeneous network derived
from a microporous carbonate dataset. The same base case is chosen for both networks
(Table 4.1). At ageing, we uniformly distribute the water-wet and oil-wet advancing
contact angles as ππ,π€π€π[πππ , 90Β°[ and ππ,ππ€π[120Β°, 180Β°], respectively. Note that the
value of πππ = 30β° corresponds to an initially water-wet rock.
πΊππ 0 π½ππ 30β°
πππ 0.5
Wettability distribution AW
Table 4.1: Base case parameters for Scenario 1 simulations.
59
4.3.1 Berea sandstone network
a) Primary drainage
In the base case, the primary drainage (PD) process is continued until the irreducible
water saturation is reached at πππππ₯ = 22πππ (Figure 4.2). As shown in Figure 4.3, all
invaded pores are sufficiently water-wet (πππ <π
2β πΎ) to hold water in the corners,
which forms the irreducible water saturation ππ€π < 0.01. In a second case, oil invasion
is stopped at prescribed initial water saturation ππ€π = 0.3 (πππππ₯ = 5πππ). As expected,
water is mainly left behind in the smallest pores, for which the ππππ‘ππ¦ are highest.
Figure 4.2: Primary drainage ππ curve for the Berea network.
Figure 4.3: Pore occupancies for the Berea network shown on the pore size distribution
following PD to different ππ€π values.
b) Ageing
The different wettability distributions (MWL, MWS, FW and AW) at fow = 0.5 resulting
after PD are plotted in Figure 4.4 on the volumetric pore size distributions. In the MWL
(resp. MWS) distribution, oil-wet pores are evidently the largest (respectively smallest).
The FW distribution results in half the pores being oil-wet for each bin size. On the other
hand, AW exhibits a more complex distribution, where both pore size and shape
determine the wettability distribution. For instance, the size of the largest pores (Rins β₯
60
41ΞΌm) prevents them from having sufficient curvature for the water film to break, thus
leaving them water-wet. Moreover, few small pores are found to be oil-wet. Indeed, the
smallest pores are mostly characterised by 3-cornered star shapes (Figure 3.9), and as
previously stated (Section 4.2), pores with smaller number of corners (reduced
roughness) are less likely to be oil-wet.
Figure 4.4: Different wettability distributions shown on the pore size distributions for
the Berea network at πππ€ = 0.5, established after PD for the base case; red: oil-wet,
blue: water-wet. Darker blue (respectively red) indicates stronger water- (respectively
oil) -wetness.
c) Waterflood
Effect of wettability distribution
The effect of varying the wettability distribution on the residual oil saturation, πππ, was
found to be small (Figure 4.5(a)). This behaviour could be explained by the homogeneity
of the sandstone structure, where pores are connected to one another regardless of
size. We will show below that the effect of the wettability distributions on πππ is much
more profound for the heterogeneous carbonate network (Section 4.3.2(c), Figure 4.15),
whose topology is much more complex. On the other hand, from the pore occupancies
at πππ€ = 0.5 (Figure 4.6) it is clear that the residual oil distribution (unlike the residual
oil saturation) is affected by the wettability distribution. In all cases, we find that the
residual oil is mostly present in oil-wet pores, as expected. A few oil layers formed near
the end of the waterflood process, mainly in the largest oil-wet pores, whose layers
61
formation ππππ‘ππ¦ are highest. The effect of the presence of oil layers on πππ is shown for
the AW distribution in Figure 4.5(b). The βwith layersβ case corresponds to the base case
where oil layers formation is enabled. On the contrary, the oil layers formation was
disabled in the βno layersβ case. As expected, oil layers allow oil to drain to lower πππ
when πππ€ is sufficiently large (for this network πππ€ β₯ 0.5) for a spanning (percolating)
oil-wet pathway to be formed.
Figure 4.5: Waterflood residual oil saturations as a function of oil-wet fractions for the
Berea network (a) for the different wettability distributions, and (b) for the AW
distribution, with and without oil layers.
Figure 4.6: Pore occupancies at the end of the waterflood for the different wettability
distributions shown on the pore size distribution for the Berea network at πππ€ = 0.5.
62
Effect of initial water saturation
A sensitivity study to ππ€π was conducted for the AW distribution. Note that because πππ€
is defined as a fraction of all pores, the maximum value that it can take is 1 β ππ€π, since
the remaining fraction ππ€π is water-filled and water-wet. This explains the different
curve endpoints for the different ππ€π in Figure 4.7.
Figure 4.7(a) shows that at πππ€ = 0.5, with ππ€π increasing from 0 to 0.1, i.e. strongly
decreasing πππππ₯ (see Figure 4.2), πππ slightly increases. This is mainly due to the fact
that the oil layer formation is inhibited at relatively low πππππ₯. The trend is amplified for
higher πππ€ = 0.8, for which the oil layers are more abundant, thus more effective in
maintaining the oil connectivity. Further increasing ππ€π leads to monotonically
decreasing πππ, as the oil saturation before waterflood, 1 β ππ€π, decreases.
Nevertheless, the effect of ππ€π on the πππ curves is considered to be small compared to
the heterogeneous carbonate case (Figure 4.16).
Figure 4.7: Waterflood residual oil saturations for the AW distribution in the Berea
network as a function of (a) ππ€π and (b) πππ€.
4.3.2 Carbonate network
a) Primary drainage
As for the Berea network, the primary drainage (PD) process is continued until the
irreducible water saturation, approximately ππ€π = 0, is reached at a relatively high
πππππ₯ = 48πππ (Figure 4.8), due to the small size of the micropores. At ππ€π = 0.3,
remaining water resides mainly in the micropores, since their ππππ‘ππ¦ are highest (Figure
4.9). Moreover, as for the Berea network (Figure 4.3), a small fraction of larger not-yet-
accessible pores still retain water.
63
Figure 4.8: Primary drainage ππ curve for the carbonate network.
Figure 4.9: Pore occupancies for the carbonate network shown on the pore size
distribution following PD to different ππ€π values.
b) Ageing
Following ageing, the resulting wettability distributions within the carbonate network
are shown in Figure 4.10. When comparing them with those for the Berea network in
Figure 4.4, it can be seen that the MWL, MWS and FW distributions are (by definition)
qualitatively quite similar. The AW distributions are similar as well, but only for the
largest pores, as that remain water-wet. However, the striking difference for AW occurs
in the smallest pores. Indeed, for the carbonate network, unlike for the Berea, 85% of
the micropores are oil-wet, since their small sizes provide sufficient curvature for the
water films to break for most of their shapes.
64
Figure 4.10: Different wettability distributions shown on the pore size distributions for
the carbonate network at πππ€ = 0.5, established after PD for the base case; red: oil-
wet, blue: water-wet.
c) Waterflood
Effect of wettability distribution
The wettability distribution significantly affects the petrophysical properties for the
carbonate network (Figure 4.11). Indeed, the residual oil varies greatly, as πππ decreases
from 0.7 for MWL to 0.24 for MWS. In addition, ππ curves are found to be dissimilar as a
result of the different pore-filling sequences. With regard to relative permeabilities, πΎπ ,
they are highly sensitive to the fluid saturations; hence we are not drawing any specific
conclusions except that they are at some extent affected by the wettability distribution
(Figure 4.11(c)). Since the relative permeabilities of water, πΎππ€, are too low to be
actually seen, we derived the fractional flow of water, πΉπ€, from the πΎπ curves, computed
assuming identical phase viscosities as:
πΉπ€ =
1
1 +πΎππ
πΎππ€
(4.3)
πΉπ€ appears in the Buckley-Leveret equation when modelling water flooding at the
continuum-scale and it shows here that water breakthrough (πΎππ€ > 0) occurs at very
different water saturations for the different wettability distributions (Figure 4.11(d)).
65
Figure 4.11: (a) ππ curves, (b) enlarged ππ curves (red box), (c) πΎπ curves and (d)
fractional flow of water, πΉπ€, curves after waterflood for the different wettability
distributions at πππ€ = 0.5 for the carbonate network.
The pore occupancies at the end of the waterflood are presented in Figure 4.12 for all
four wettability distributions. Note that, in general, water would start filling water-wet
pores (from small to large), then oil-wet pores (from large to small), unless this is
prevented by lack of accessibility and trapping. Since the fine scale pores provide the
overall connectivity for the disconnected coarse scale pores (Section 3.3.2), two
interesting limiting cases stand out, MWL and MWS.
In the MWL distribution, a fraction of the fine scale pores are water-wet, thus first filled
with water. However, these water-filled pores now block the escape of oil from the
coarse scale pores, thus leaving much oil trapped in the largest pores, as well as in
intermediate-sized pores and some micropores. The capillary pressure curve for this
case (see Figure 4.11(a) and (b)), confirms that almost exclusively small, i.e. water-wet,
pores are invaded. Hence, trapping in the MWL case is high compared to the other
distributions.
Conversely, in the MWS case, water first starts filling the larger water-wet pores. Note
that all pores have corner wetting films (see Figure 4.8). Therefore, all pores are
accessible to the invading water, even though the coarse scale network is disconnected.
66
Water invasion then continues in the smaller oil-wet pores as a drainage process. The
corresponding jump in invaded pore size is translated into a large drop in ππ over a small
saturation range in the capillary pressure curve (Figure 4.11(b)). By leaving the
micropores to be filled at the end of the process, MWS has the lowest trapping of all
wettability distributions.
The pore occupancy for the AW distribution (Figure 4.12) is quite similar to that for
MWS, in contrast to the occupancies for the Berea network, while the occupancy for FW
exhibits behaviour intermediate to MWS and MWL.
Figure 4.12: Pore occupancies at the end of the waterflood for the different wettability
distributions shown on the pore size distribution for the carbonate network at πππ€ =
0.5.
The oil phase connectivity, translated into ππππΜ , is shown in Figure 4.13 as a function of
the water saturation during waterflood for the different wettability distributions. It
confirms that the residual oil saturation is controlled by the rate of trapping of the oil
phase during water injection. In fact, MWL, FW, AW and MWS systems exhibited slower
decrease in ππππΜ , in order, which resulted in decreasing residual oil saturations, in order.
67
Figure 4.13: Evolution of the oil phase connectivity (normalised Euler number) during
waterflood for the different wettability distributions at πππ€ = 0.5 in the carbonate
network.
Effect of oil-wet fraction
We now consider the AW distribution, and study the impact of changing the fraction of
oil-wet pores, πππ€, on the petrophysical properties. As expected, ππ curves are lower for
larger fractions of oil-wet pores (Figure 4.14). Considering the residual oil saturation, the
trend is visibly monotonic, as πππ decreases with πππ€. We note as well that the switch
from fully water-wet (πππ€ = 0) to less water-wet (πππ€ = 0.2) leads to a large reduction
in πππ, as a fraction of the micropores become oil-wet. However, further changing to
fully oil-wet has a relatively small effect. The impact of the presence of oil layers on πππ
shown in Figure 4.15(b) is similar to that seen in the Berea network (Figure 4.5(b)).
Figure 4.14: ππ curves after waterflood for different oil-wet fractions for the carbonate
network.
68
The results of this sensitivity study are summarised in Figure 4.15(a), where both the
wettability distribution and πππ€ are varied. Unlike for the Berea sandstone (see Figure
4.5(a)), the πππ values are very different for the various wettability distributions in this
microporous carbonate. The MWS and MWL distributions clearly form the limiting
boundaries for the residual oil saturations. Indeed, the best recovery is exhibited by
MWS, while MWL shows the worst recovery. On the other hand, recoveries for the
developed AW distribution and the FW distribution lie between the two extreme cases,
but they still differ significantly from each other.
Figure 4.15: Waterflood residual oil saturations as a function of oil-wet fractions for the
carbonate network (a) for the different wettability distributions, and (b) for the AW
distribution, with and without oil layers.
Effect of initial water saturation
πππ curves are plotted in Figure 4.16 for different ππ€π values for the AW distribution. The
trends are similar to those observed in the Berea network (Figure 4.7) but the variations
and the differences between the curves for varying πππ€ are much larger. Actually, while
increasing ππ€π, added to the fact that fewer oil layers form, more water is retained in
the micropores at the start of the waterflood, which in turn blocks the escape of oil in
the bigger pores due to the particular connectivity of the carbonate network.
To examine the structure of the residual oil, we show in Figure 4.17 the pore occupancies
at the end of the water flood for different ππ€π values at πππ€ = 0.5. Since the initial water
resides mostly in the micropores (see Figure 4.9), the nature of the residual oil changes
with higher ππ€π, as oil is increasingly trapped in intermediate-sized pores and less so in
69
micropores. Besides, as previously stated, oil layers are less likely to develop at
lower πππππ₯, i.e. at higher ππ€π.
Figure 4.16: Waterflood residual oil saturation for the AW distribution in the carbonate
network as a function of (a) ππ€π and (b) πππ€.
Figure 4.17: Pore occupancies at the end of the waterflood for different ππ€π values
shown on the pore size distribution for the carbonate network at πππ€ = 0.5.
Note that by keeping πππ€ constant (equal to 0.5) and increasing ππ€π (decreasing πππππ₯)
(Figure 4.16), the corresponding Πππππ‘ value is decreasing. This may not be realistic since
Πππππ‘ is assumed to be an intrinsic property of the rock mineral and water film sub-
system. Indeed, the βrightβ way would be to keep Πππππ‘ constant, increase ππ€π, and
consequently obtain a lower πππ€ (see Figure 4.18, moving along vertical lines). For
instance, if Πππππ‘ = 48.33πππ was chosen such that πππ€ = 0.5 for πwi = 0 ( Pcmax =
47.8πππ), changing ππ€π to 0.1 would reduce πππππ₯ to 27.8kPa. The latter value would
be below the critical capillary pressure ππβ of all pores, keeping all water films intact,
thus πππ€ = 0.
70
Figure 4.18: Contour chart describing the relationship between Πππππ‘, ππ€π and πππ€.
4.4 Discussion
First, note that since we are lacking real ππ€π (or ππmax) data, zero initial water saturation,
Sπ€π = 0, was chosen as a base case to highlight the impact that the wettability of
micropores has on recovery. In fact, oil presence in micropores has been reported in
many real carbonate rocks, especially towards the top of the oil column. This could
happen if the reservoir has a sufficiently large oil column such that a high capillary
pressure is reached in the upper oil column, or if some pores have somehow undergone
a change in size (e.g. by means of dissolution/cementation) or in wettability (e.g. due to
polar species in the oil) over geological time.
Wettability has long played the role of a sort of βtuning parameterβ in simulations,
although in actual calculations it is the consequent petrophysical function (ππ and πΎππ)
that is actually used or varied; no βnumber for wettabilityβ appears in any oil
displacement calculation. However, there is strong evidence from pore network
modelling (supported again by the results presented here) that the actual assumptions
on the wetting distribution do strongly influence these petrophysical functions as well
as the πππ value. Thus, if we had some prior knowledge of these wetting patterns, then
we might be able to forward model the petrophysical functions. Working
petrophysicists and core analysts in the oil industry do have views on the physical forms
of these wetting patterns. Therefore, we suggest building up βType diagramsβ of
wetting patterns in carbonate rocks by consideration of the possible mechanisms
involved, such as those incorporated in the developed AW model. The patterns
generated can then be presented to petrophysics experts who will recognise the most
71
likely physically realistic patterns. This approach is used in other industries and is known
as expert elicitation (Curtis and Wood, 2005). At best such a scenario might generate in
a systematic manner prior probabilities of certain wetting patterns for realistic
carbonates. We believe that this is the only way that a combination of flow physics,
forward modelling and industry expertise/knowledge can be combined to make some
advances in modelling complex mixed-wet systems at the pore level.
4.5 Conclusions
We developed a physically-based wettability alteration scenario, dependent on both
pore size and shape, which incorporates a plausible view of the wetting change
mechanism. The scenario qualitatively reproduced a pattern of wettability observed in
microporous carbonates through high-resolution imaging, where anhedral (curved,
rough and poorly formed) faces become preferentially oil-wet. We implemented the
scenario in a two-phase quasi-static pore network model which involves a variety of pore
shapes. We considered as input two pore networks with very different levels of
complexity in their pore structure, (a) a fairly homogeneous connected Berea sandstone
network with a relatively narrow range of pore sizes, and (b) a heterogeneous two-scale
carbonate network whose coarse scale pores were not connected, but where the fine
scale pores provided overall connectivity to the network. We considered the widest
possible range of wettability distributions including the newly developed Altered-Wet
(AW) distribution, along with the commonly used MWL, MWS and FW distributions. This
resulted in a correspondingly wide range of outcomes in terms of pore occupancies, ππ
and πΎπ curves and πππ values.
After ageing was carried out, the AW distribution resulted in the largest pores being
water-wet, as their large size prevented them from having sufficient curvature for the
water film to collapse. Conversely, owing to their tiny size, most of the carbonate
networkβs micropores were found to be oil-wet, provided that they were invaded by oil
during primary drainage. Yet, the AW distribution was still distinct from the MWS, as
well as from any of the other common wettability distributions. Following
waterflooding, we showed from the pore occupancies that the specific wettability
distribution affects the structure of the residual oil. In addition, wettability proved to
have some effect on the residual oil saturation (πππ) for the Berea network, but it had a
much larger impact for the multiscale carbonate network. Indeed, since the connectivity
72
of this network is mainly driven by pore size, the MWS and MWL distributions formed
limiting cases for the πππ values, with the recoveries for the AW and FW distributions
lying between these two extremes. The MWL case exhibited by far the lowest oil
recovery since the first-filled water-wet micropores blocked the escape of oil from the
larger pores. Conversely, the MWS distribution showed the best recovery as the oil-wet
micropores are left to be filled with water at the end of the waterflood. In addition, the
relative permeability curves for the carbonate network were very sensitive to the
chosen wettability distribution. This was emphasised by the corresponding fractional
flow curves, which showed very different water breakthrough saturations. Furthermore,
we demonstrated that oil layers did indeed allow oil to drain to lower πππ, but only at a
sufficiently high πππ€. Moreover, by increasing ππ€π, we first observed higher πππ, due to
fewer oil layers being formed, and then lower πππ because of a decreasing oil saturation
before waterflood. We proved that these trends are amplified at higher πππ€ values
where more oil layers form, as well as in the carbonate network where the effect of the
networkβs particular connectivity contributes notably. Indeed, the increased volumes of
connate water left behind in the micropores consequently trap the oil in the larger
pores.
73
Chapter 5 : Scenario 2 β Wettability alteration starting during primary
drainage
5.1 Introduction
In the previous chapter, we examined a physically-based wettability alteration model,
referred to as Altered-Wet (AW), and assessed the consequences of wettability
distributions in pore network models. Indeed, we described in a simple manner the rule-
based wetting change mechanism that strictly occurs following primary drainage, due
to the surface adsorption of such species as asphaltenes subsequent to ageing in crude
oil. This corresponds to the traditional approach that mimics the 3-stage process
experienced by an initially water-wet reservoir: primary drainage, ageing and
waterflood. Interestingly, we demonstrated the importance of the wettability of
micropores on oil recovery, but only if they are actually invaded by oil. Although this
may seem unlikely due to the very high capillary pressures needed to invade such tiny
pores, oil has actually been detected within micropores in carbonates (Al-Yousef et al.,
1995, Clerke, 2009, Knackstedt et al., 2011, Fung et al., 2011, Clerke et al., 2014, Dodd
et al., 2014) which may be explained by a wettability change that occurred progressively
over geological time.
In this chapter, we present a more mechanistic physically-plausible model for the initial
stages of wetting change from water-wet to more intermediate-wet conditions, which
may occur during primary drainage. We implement the model in the quasi-static pore
network model described in Section 3.2, to explain and numerically simulate the
wettability alteration mechanism suggested by Bennett et al. (2004), described in
Section 2.1.1, involving smaller polar compounds from the oil. We have added a
βscaledβ time-dependency to the common oil invasion-percolation algorithm to be able
to incorporate a time-dependent transport model for polar compounds. The time scaling
referred to incorporates the balance between the oil invasion/migration timescale and
the timescale of diffusion of small polar species and their adsorption which triggers a
wetting change. The physical and chemical processes in this model will affect the final
phase distributions and initial water saturations in the oil column after oil migration.
Particularly, the model provides a clear and precise mechanism of how oil can migrate
into micropores, without necessarily reaching the excessively high capillary pressures
74
that would be required for oil invasion into strongly water-wet micropores.
Subsequently, we model the full βageingβ change associated with asphaltenes and the
resulting oil-wet conditions. In Section 5.2, details are given of how small polar non-
hydrocarbon molecules diffuse from the oil into and through the water phase within the
pore network, thus leading to wettability alteration which can progress as the primary
drainage process occurs. In Section 5.3, we present the resulting fluid distributions at
the pore level (pore occupancies) and the consequences of these changes on the phase
saturation profiles within the oil column during primary drainage, and their effect on
subsequent imbibition. These calculations are based on the two networks with distinct
pore-structures introduced in Section 3.3: the homogeneous Berea network and the
heterogeneous multi-scale carbonate network.
5.2 Model Description
The model starts with initially water-filled and perfectly water-wet pores, the initial
oil/water contact angle, π = 0Β°. Commonly in quasi-static pore network modelling
simulators (Chapter 4), oil invasion occurs pore by pore at discrete invasion percolation
(IP) events. These are assumed to happen instantaneously, hence time is not explicitly
taken into consideration. In this chapter, we separate these invasion events in time,
based on an assumed flow rate π, corresponding to a charge time for an oil reservoir by
oil migration.
In fact, we assume that the invading volumes of oil, ππππ[π3] linearly increase over time:
Vπππ(π‘) = π β π‘ , where π‘[s] is the migration time; π[π3π β1] is the flow of hydrocarbons
invading a reservoir rock. π is chosen low enough to remain in the capillary-dominated
regime by satisfying Equation (3.1) using π = π/π΄πππππ‘, with π΄πππππ‘ the total cross-
sectional area at the system inlet. We effectively replace a discrete process done in steps
by a continuous process described by the straight line in Figure 5.1. Note that the
individual volumes of the invaded pores are expected to decrease in time during primary
drainage, since oil generally fills the water-wet pores from large to small in size.
At time π‘π during instantaneous oil invasion of pore π, we consider the polar compounds
to be transported through the oil phase. The details of this process are provided in
Section 5.2.1. Additionally, during the period of time π‘π+1 β π‘π separating two successive
pore invasion events, a diffusion/adsorption model for polar compounds is applied using
constant discrete time steps, βπ‘ππ (Figure 5.1). In fact, we assume that the polar
75
compounds diffuse to - and adsorb in - the oil-filled pores, as well as in the water-filled
pores due to their high solubility in water (Bennett et al., 2004), as detailed in Section
5.2.2. The initial conditions for the diffusion/adsorption process are related to the
oil/water configuration and their relative concentrations at time π‘π.
Figure 5.1: Illustration of the separation in time of the oil invasion events, where the
cumulative volume of pores invaded linearly increases over time based on the assumed
flow rate, π; and the incorporation of a diffusion/adsorption model for polar
compounds using discrete time steps, βπ‘ππ , during the period of time π‘π+1 β
π‘π separating two successive pore invasion events.
Note that some particular cases may arise as a consequence of the assumptions that we
have made:
The period π‘π+1 β π‘π is known based on the next anticipated invasion. Hence, if a
spontaneous invasion, usually of a smaller pore, takes place as a result of contact
angle changes, π‘π+1 will be smaller and the period π‘π+1 β π‘π will be shorter.
Therefore, we may already have diffused for too long.
When the predefined maximum capillary pressure, πππππ₯, is reached and no
more invasions are anticipated, the diffusion/adsorption model is carried out
and βπ‘ππ incremented until a spontaneous invasion becomes available due to
contact angle change.
The period π‘π+1 β π‘π is generally not a multiple of βπ‘ππ . Thus, we assume that the
diffusion/adsorption model stops before the next invasion at time π‘π+1 is
exceeded when incrementing βπ‘ππ .
76
In the particular case of a very high flow rate where π‘π+1 β π‘π < βπ‘ππ , the two
successive pore invasions at π‘π and π‘π+1 occur and the diffusion step is delayed
until the above condition does no longer hold or until the predefined πππππ₯ is
reached and no more invasions are anticipated.
5.2.1 Transport through oil invasion
Throughout this section, we will consider a single pore π and define πΆππ(t) [
πππΏβ ] and
πΆππ€(t) [
πππΏβ ] as the mobile concentrations of polar compounds in the oil phase and
water phase, respectively, and π€i(t) [ππ
π2β ] as the corresponding adsorption level of
polar compounds per unit area in pore π at time π‘. Note that we assume perfect mixing
within each phase.
If pore π is water-wet and has corners, water remains in the pore corners, as well as in a
thin film lining the pore wall following oil invasion (Kovscek et al., 1993). When polar
compounds are transported into the oil-phase of pore π, either through oil invasion or
through diffusion, we assume that they instantaneously partition between the oil and
water phases within the same pore, since they are highly soluble in water, and that they
instantly adsorb onto the surface from the water-phase. These are reasonable
assumptions given the timescales of diffusion compared to the slower migration of oil.
An illustration of the partitioning and adsorption within an angular oil-filled pore is
shown in Figure 5.2. We also assume that the concentration of polar compounds in the
water-phase, πΆππ€(π‘) , is linearly related to the oil-phase concentration, πΆππ
(π‘), as
follows:
πΆππ€
(π‘) = ππΆππ(π‘) , βπ‘ > 0 (5.1)
where π is the partitioning coefficient, input to the model (0 < π < 1).
During oil invasion of pore π at time π‘π, as illustrated in Figure 5.2, we consider the polar
compounds to be carried by the oil phase. Hence, we assume that πΆππ(π‘π) is equal to the
concentration at the oil invasion front, computed as an average of the oil phase
concentrations in the connecting oil-filled pores:
77
πΆππ
(π‘π) =β πΆππ
(π‘π) πππ
π§ππ=1
β πππ
π§ππ=1
(5.2)
We define πππ and πππ€
as the volumes of the oil phase and water phase of pore π,
respectively, satisfying πππ+ πππ€
= ππ. Besides, π§π is the number of oil-filled pores
adjacent to pore π, π§π β€ π§, where π§ is the total number of pores connected to pore π; π§
is equal to 2 for a bond, corresponding to its two connecting nodes (π§ = 1 for a
boundary bond) and π§ β₯ 1 for a node, corresponding to its coordination number
(number of neighbouring bonds).
The oil invasion of pore π at time π‘π results in a gain in the mass of polar compounds in
the oil-phase in the network, equal to πΆππ(π‘π) β πππ
. In order to satisfy the mass balance
of polar compounds, we assume that this extra mass is supplied at the inlet. Besides, if
prior to invasion, pore π had a non-zero concentration of polar compounds, πΆππ€(π‘π
β), we
assume that this concentration is unchanged immediately after oil invasion, i.e.
πΆππ€(π‘π) = πΆππ€
(π‘πβ), resulting in a loss of some mass of polar compounds from the water
phase due to the decrease in its volume, equal to πΆππ€(π‘π) β πππ
. This mass is assumed to
be expelled through the outlet. The total mass within the pore at time π‘π, πi(π‘π), is equal
to πΆππ(π‘π) β πππ
+ πΆππ€(π‘π) β πππ€
+ π€π(π‘π) β ππ΄π, where ππ΄π [π2] is the total surface area
of pore π. Afterwards, at π‘π+, this mass is supposed to instantaneously distribute
between the phases within the same pore and to adsorb onto the surface in accordance
with a Langmuir isotherm as follows:
π€π(π‘π
+) = π€πππ₯
πΎπΆππ€(π‘π
+)
1 + πΎπΆππ€(π‘ + βπ‘
ππ )
πi(π‘π+) = πΆππ
(π‘π+) β πππ
+ πΆππ€(π‘π
+) β πππ€+ π€π(π‘π
+) β ππ΄π = πi(π‘π)
(5.3)
where π€πππ₯ [ππ
π2β ] and πΎ [πΏππβ ] are the Langmuir maximum adsorption level per
unit area and adsorption constant, respectively.
By combining Equations (5.1) and (5.3), we obtain a single quadratic equation that we
solve for πΆππ(π‘π
+), knowing πΆππ(π‘π), πΆππ€
(π‘π), and π€(π‘π):
πΎπ. πΆππ(π‘π
+)2 + (1 +πΎπ
πππ+ ππππ€
(π€πππ₯. ππ΄π β πi(π‘π))) . πΆππ(π‘π
+) βπi(π‘π)
πππ+ ππππ€
= 0 (5.4)
78
Afterwards, πΆππ€(π‘π
+) and π€π(π‘π+) are directly derived from equations (5.1) and (5.3),
respectively.
Figure 5.2: Illustration of the transport of polar compounds within a pore π with
triangular cross-section during oil invasion (at time π‘π); as well as their distribution
between the phases within the same pore and their adsorption onto the surface, which
are assumed to happen right after invasion (at time π‘π+). Note that πΆππ
and πΆππ€are the
mobile concentrations of polar compounds in the oil phase and water phase,
respectively, and π€π is the corresponding adsorption level of polar compounds per unit
area.
5.2.2 Transport through diffusion
As previously stated, polar compounds diffuse into both oil-filled and water-filled pores
during the period of time separating two successive pore invasions at discrete time-
steps βπ‘ππ .
a) Mass balance equations
Oil-filled pores
The material balance of the polar compounds within pore π is described by the one-
dimensional diffusion-adsorption equation in the following discretized from:
βπΆππ
βπ‘ππ =
πΆππ(π‘ + βπ‘
ππ ) β πΆππ(π‘)
βπ‘ππ =
1
πππβπ‘
ππ (βπππβ βπππ₯π
)
βπΆππ€
βπ‘ππ =
πΆππ€(π‘ + βπ‘
ππ ) β πΆππ€(π‘)
βπ‘ππ =
1
πππ€βπ‘
ππ (βπππ₯πβ βπππ
)
(5.5)
79
where βπππ, βπππ₯π
and βπππ [ππ] are the masses of polar compounds that diffused
into the bulk phase (here oil), that exchanged between the oil and the water phases and
that adsorbed onto the surface, respectively, in pore π during time increment βπ‘ππ [π ];
Additionally, we assume that the diffusion occurs only through the bulk phase,
neglecting any mass exchanged with the neighbouring pores through the corner water
phase. Since βπππ₯π is the same mass exchanged between oil and water, the two
Equations in (5.5) lead to a single equation:
πππ
(πΆππ(π‘ + βπ‘
ππ ) β πΆππ(π‘)) + πππ€
(πΆππ€(π‘ + βπ‘
ππ ) β πΆππ€(π‘)) = βπππ
β βπππ (5.6)
Water-filled pores
Additionally, due to their supposedly high water solubility, polar compounds are
assumed to diffuse into the water-filled pores where they also adsorb onto the surface.
Since the pore only contains a water phase, the material balance for the polar
compounds is simply provided by the following equation:
βπΆππ€
βπ‘ππ =
πΆππ€(π‘ + βπ‘
ππ ) β πΆππ€(π‘)
βπ‘ππ =
1
ππβπ‘ππ (βπππ
β βπππ) (5.7)
b) Diffusion model
We apply a discretized form of Fickβs diffusion equations for the diffusion of polar
compounds within pore π (Equation (5.8)), with the diffusion front only advancing to the
directly adjacent pores ahead at each time-step βπ‘ππ .
βπππ
= βπ‘ππ β π½ππ . min (π΄π, π΄π)
π§
π=1
π½ππ = βπ·πΆπ(π‘) β πΆπ(π‘)
πΏππ
(5.8)
where π§ is the number of pores connected to pore π (as defined above); π½ππ [ππ
π2π β ] is
the diffusion flux from pore π to pore π, which occurs across the minimum bulk phase
cross-sectional area, π΄ [π2], between the two adjacent pores; πΏππ [π] is the distance
between the centre of pore π and centre of pore π; π· [ππ2
π β ] is the diffusion coefficient;
Note that at fixed βrealisticβ diffusion constant π·, we should carefully choose a low βπ‘ππ
80
in order to avoid instabilities that arise from the discretised diffusion model. Indeed,
unless every pore satisfies the condition below, mass conservation fails:
βπππ
+ ππππ’ππ(π‘) β₯ 0 (5.9)
where ππ’ππ corresponds to the poreβs bulk phase, through which the diffusion occurs;
and ππππ’ππ(π‘) is the mass at the beginning of the time-step in the bulk phase, defined as
ππππ’ππ(π‘) = πΆπππ’ππ
(π‘)ππππ’ππ.
As for the boundary conditions, the concentration at the inlet is taken as constant over
time, equal to πΆ0. At the outlet, a βno flow" boundary condition is assumed. The
boundary conditions are translated into the following equations:
πΆππ(π‘) = πΆ0 , βπ‘ > 0, βπ βnotionalβ inlet node
πΆππ(π‘) = πΆ(πβ1)π
(π‘) , βπ‘ > 0, βπ βnotionalβ outlet node (5.10)
where (π β 1) is the outlet bond connected to the βnotionalβ outlet node π. The
βnotionalβ boundary nodes are illustrated in Figure 3.8.
We model the diffusion between two pores sharing the same bulk phase using the bulk
phase concentrations i.e. the diffusion from oil- to oil-filled, and water- to water-filled
pores is computed in Equation (5.8) using πΆππ(π‘) β πΆππ
(π‘) and πΆππ€(π‘) β
πΆππ€(π‘) respectively (Figure 5.3(a)). For the case of cross-phase diffusion i.e. from water-
to oil-filled pores and vice-versa, we use the difference in water concentrations:
πΆππ€(π‘) β πΆππ€
(π‘). Indeed, we assume that the polar compounds partition first from the
oil to the water phase at the interface between two adjacent pores, then they diffuse
within the water phase (Figure 5.3(b)). In this particular case, πΏππ is taken as only half the
length of the water-filled pore.
81
Figure 5.3: Illustration of the diffusion process between (a) two adjacent pores sharing
the same bulk (oil) phase and (b) an oil-filled pore adjacent to a water-filled pore
(cross-phase diffusion) with the partitioning of polar compounds from the oil to the
water phase at the interface and their diffusion within the water phase. Note that
πΆππ and πΆππ€
are the mobile concentrations of polar compounds in the oil phase and
water phase, respectively; π½ππ and πΏππ are the diffusion flux and length, respectively,
from pore π to pore π. Note as well the colour key provided in Figure 5.2.
c) Adsorption model
The adsorption of polar compounds is assumed to occur from the water phase
instantaneously. However, since the polar compounds concentration changes through
diffusion at discrete time-steps βπ‘ππ , adsorption is only computed at each βπ‘
ππ . It is
described by a Langmuir isotherm which has the following form:
βπππ= ππ΄π( π€π(π‘ + βπ‘
ππ ) β π€π(π‘))
π€π(π‘ + βπ‘ππ ) = π€πππ₯
πΎπΆππ€(π‘ + βπ‘
ππ )
1 + πΎπΆππ€(π‘ + βπ‘
ππ )
(5.11)
d) Equations summary
Oil-filled pores
To summarise, in an oil-filled pore π at π‘ + βπ‘, we have βπππ= βπππ
(πΆπ(π‘)) and
βπππ= βπππ
(π€π(π‘) , π€π(π‘ + βπ‘ππ ), πΆππ€
(π‘ + βπ‘ππ )). Hence, Knowing π€π(π‘) , πΆππ
(π‘) and
πΆππ€(π‘) from the previous time-step, we combine Equations (5.6), (5.8) and (5.11) to
obtain a single quadratic equation that we solve for πΆππ(π‘ + βπ‘
ππ ):
82
πΎπ. πΆππ
(π‘ + βπ‘ππ )2 + (1 β πΎππ΅ +
π€πππ₯πΎπ. ππ΄π
πππ+ ππππ€
) . πΆππ(π‘ + βπ‘
ππ ) β π΅ = 0 (5.12)
where π΅ = πΆππ(π‘) +
βπππ+π€π(π‘).ππ΄π
πππ+ππππ€
.
Afterwards, πΆππ€(π‘ + βπ‘
ππ ) and π€π(π‘ + βπ‘ππ ) are directly derived from equations (5.1) and
(5.11), respectively. Since the concentration of polar compounds in the inlet is constant,
equal to πΆ0, πΆππ and πΆππ€
increase from 0 to πΆ0 and ππΆ0, respectively, following the
transport model.
We will show below that both the water present in corners and present as a film may
collapse in some pores due to polar compounds adsorption; but when this happens,
adsorption is assumed to carry on from the oil phase using Equation (5.12), which
coincides with Equation (5.13) below for water-filled pores with swapped phases and an
adsorption constant equal to πΎΓπ.
Water-filled pores
Similarly, for a water-filled pore π, we combine equations (5.7), (5.8) and (5.11), and
solve for the unknown πΆππ€(π‘ + βπ‘
ππ ):
πΎ. πΆππ€
(π‘ + βπ‘ππ )2 + (1 β πΎπ΅ +
π€πππ₯πΎ. ππ΄π
ππ) . πΆππ€
(π‘ + βπ‘ππ ) β π΅ = 0 (5.13)
where π΅ = πΆππ€(π‘) +
βπππ+π€π(π‘).ππ΄π
ππ.
π€π(π‘ + βπ‘ππ ) is directly derived from equation (5.11). Besides, πΆππ€
increases from 0 to ππΆ0
following the transport model.
5.2.3 Wettability alteration
Due to the polar species adsorption, the pore surface undergoes a wettability alteration
(Bennett et al., 2004). We simply assume that the cosine of the contact angle changes
as a linear function of the adsorption level of polar compounds:
cos ππ = 1 β (1 β π½)
π€π
π€πππ₯ (5.14)
While π€π = 0, the initial contact angle remains unchanged i.e. cos ππ = 1. Note that π½ is
an input parameter ranging between 0 and 1, and corresponds to the limiting contact
83
angle value where π€π = π€πππ₯. However, in accordance with the Langmuir adsorption
isotherm Equation (5.11), the actual maximum value π€πππ₯ _πππ‘π’ππ that π€π can reach as the
transport model is carried out is:
π€πππ₯ _πππ‘π’ππ = π€πππ₯
πΎππΆ0
1 + πΎππΆ0 (5.15)
Hence, by combining Equations (5.14) and (5.15), cos ππ decreases during the simulation
from 1 to a minimum value, related to a maximum contact angle ππππ₯:
πππ ππππ₯ =
1 + π½πΎππΆ0
1 + πΎππΆ0 (5.16)
Consequently, π½ can be defined as π½ = πππ ππππ₯ (πΎ β +β) at finite πΆ0 (or
as πππ ππππ₯ (πΆ0 β +β) at finite πΎ). We identify distinct effects of the wetting change
in the oil-filled and water-filled pores, as described below.
a) Oil-filled pores
The wetting change in an oil-filled pore may lead to two consequences: the shrinking
(and possible collapse) of the water films in the corners, and destabilisation (and
possible collapse) of the thin water films that coat the pore walls.
Corner water shrinking
Due to the uniform contact angle change in any angular pore π, water in the pore corners
shrinks if it is connected to the outlet. Water may be completely expelled from the pore
if it satisfies the condition below:
ππ >
π
2β πΎπ (5.17)
where πΎπ is the half-angle of the angular cross-sections, as illustrated in Figure 5.4.
84
Figure 5.4: Illustration of the corner water shrinking (and possible collapse) due to the
contact angle, π, increase within a pore with triangular cross-section and half-angle πΎ.
Note the colour key provided in Figure 5.2.
A direct consequence of this mechanism is that water may get surrounded by oil in the
vicinity of the water films collapse, which generates trapping of water in the network. In
fact, the water phase connectivity may drop significantly at this stage due to large
contact angle changes.
Notice that during the corner water shrinking process in pore π, we assume for simplicity
that π€π, πΆππ and πΆππ€
are kept constant. As a consequence, if the change in the oil phase
volume within the pore is βπππ> 0, an extra mass of polar compounds, πΆππ
βπππ(1 β π),
is assumed to be supplied to the pore from the inlet.
Thin water films destabilisation
The adsorption of polar species affects the stability of thin water films by weakening the
bonds that keep them in place. Indeed, at fixed time π‘, corresponding to a fixed capillary
pressure, ππ, thin-film forces exist within the stable water film that support it. The
related pressure is the so-called disjoining pressure, Ππ. It is linked to ππ through the
augmented Young-Laplace equation (Equation (2.2)), that we have slightly modified by
incorporating an adsorption coefficient, πΌπ β [0,1]:
ππ = πΌπΠπ + πππ€ππ
(5.18)
where ππ is the equivalent wall curvature of pore π introduced in Section 4.2.
We model πΌπ as πΌπ = cos ππ . Indeed, when the surface of pore π adsorbs polar
compounds, thus increasing π€π (Equation (5.14)), πΌπ decreases from its initial value 1 (no
85
adsorption). Consequently, Ππ rises to counterbalance it (since ππ is fixed) and the water
film gets thinner (Figure 5.5). If Ππ increases enough to reach the critical disjoining
pressure Πππππ‘ i.e. ππ > πππ
β = πΌπΠππππ‘ + πππ€ππ, the film is no longer stable and collapses.
Because Πππππ‘ is an intrinsic property of the rock (depending on the fluid system and
mineralogy of the rock surface), we suppose it is uniform throughout the rock and
constant throughout the process.
Figure 5.5: Evolution of thin film stability during polar compounds adsorption
(increasing π€ i.e. decreasing πΌ = πππ π, where π€ and π are the adsorption level of polar
compounds per unit area and contact angle, respectively, linked through Equation
(5.14)) at fixed ππ, illustrated on an example of disjoining pressure isotherm Π(π).
b) Water-filled pores
As the adsorption levels increase during the transport process in a water-filled pore π
adjacent to the oil front, the contact angle increases. Consequently, the pore entry
pressure, ππππ‘ππ¦, decreases, which makes the pore more prone to oil invasion. Oil may
then spontaneously invade pore π when its wettability has changed enough for its entry
pressure, ππππ‘ππ¦, to decrease below the current capillary pressure value ππ. Notice that
this particular time of spontaneous oil invasion cannot be explicitly determined because
of the non-linearity of the relationship between ππππ‘ππ¦ and the contact angle π.
Note that at π sufficiently close to 90Β°, all the corner water films are removed if
connected to the outlet, in accordance with Equation (5.17), and all the entry pressures
are close to zero, so all invasions are possible at πππππ₯ > 0. This justifies the upper limit
of 90Β° for π (π½ = 0) introduced above in Equation (5.14), since choosing a higher contact
angle has little effect, if any, on the final ππ€π following primary drainage.
86
The simulation is stopped when no more invasions are possible at a predefined
maximum capillary pressure, πππππ₯ and the adsorption steady-state is reached. The
adsorption steady-state is defined as the condition where all the remaining accessible
and non-trapped water-filled pores have adsorbed their maximum capacity of polar
compounds, i.e. their contact angles have reached a final value equal to ππππ₯ (Equation
(5.16)). The ultimate water saturation obtained at the predefined πππππ₯ will simply be
referred to as ππ€π (corresponding to an oil saturation πππ = 1 β ππ€π), which is reached
at a final time denoted as π‘π. We refer to the newly developed model of Primary
Drainage during which a Wettability Evolution occurs as PD/WE. An example of the first
steps of the simulation is carried out on a regular 2D network and shown in Figure 5.6.
It illustrates the various processes involved in the PD/WE model.
Figure 5.6: Network-scale representation of the PD/WE model, involving the time-
dependent oil invasion coupled with the transport model for polar compounds; π‘π is the
invasion time of pore π, in accordance with the example in Figure 5.1. Oil displaces
water from the inlet (left) to the outlet (right) of a 2D regular network. Note the colour
key provided in Figure 5.2.
87
5.3 Results and discussion
The simulations are performed on the same two networks described in Section 3.3. Each
network consists of pores with different cross-sectional geometries including regular n-
cornered polygons and stars. Note that the 5-cornered star shapes are predominant in
both networks, with their corresponding half-angle πΎ ranging between 5 and 54Β°. The
latter value corresponds to the pentagon shape.
In this section, we link the pore-scale simulations to the reservoir scale. In fact, we
associate each ππ reached locally in the network with a corresponding height, β, in the
oil column:
β =
ππ
βπ. π (5.19)
where βπ = ππ€ β ππ denotes the difference between the fluid volumetric mass
densities (here chosen as 0.2 ππ/πΏ); π denotes the gravity (9.81 π/π 2).
Actually, two main effects will be demonstrated in our simulations, both of which dictate
the initial water saturation for waterflooding, ππ€π:
The effect of the contact angle, ππππ₯ on ππ€π: while πΎ, π and πΆ0 are kept constant,
π½ will be varied to induce changes in the contact angles (Equation (5.16)). We
either assign a unique π½ throughout the network, corresponding to a single ππππ₯
for all the pores, or uniformly distribute π½ among the pores, generating a range
of contact angles ππππ₯ .
The effect of the balance between oil invasion and wettability alteration on ππ€π:
while we keep the oil flow rate, π, constant, we vary the maximum adsorptive
capacity, π€πππ₯. Hence, the polar compounds adsorption level required to reach
the fixed maximum contact angle ππππ₯ is changed (Equation (5.15)). This induces
changes in the balance between the processes of oil invasion and wettability
alteration.
Note that for the latter sensitivity study at fixed ππππ₯, two limiting cases arise:
In the case of an extremely fast wettability alteration relative to flow rate, a full
wetting change within every pore precedes its invasion by oil. Hence, we can
88
simply model it as a conventional PD with an initial contact angle equal to ππππ₯.
We call this special case the Fast Wetting Boundary (FWB).
If the oil flow rate is much faster than the wetting change, all invasions are
supposed to happen instantly. The oil invasion process is carried out until the
imposed maximum capillary pressure, πππππ₯, is reached. Only after this point the
wettability alteration takes effect, starting from the inlet, resulting in subsequent
spontaneous oil invasions. We refer to this limiting case as Slow Wetting
Boundary (SWB).
5.3.1 Berea sandstone network
We simulate the above PD/WE model on the homogeneous Berea network described in
Section 3.3.1. The base case parameters of the simulations are summarised in Table 5.1.
Note that the chosen π½, πΎ, π and πΆ0 parameters correspond to ππππ₯ = 80Β° (Equation
(5.16)). Although our base case parameters have not been derived from a particular
experiment, they are physically realistic, chosen in order to scan all the possible
outcomes of our model during the sensitivity analysis that will follow.
π [π3
π β ] 5e-13
πΆ0 [ππ
πΏβ ] 500
βπ‘ππ [π ] 0.007
π 0.01
π· [ππ2
π β ] 1e-5
πΎ [πΏππβ ] 1
π€πππ₯[ππ
π2β ] 0.3
π½ 0.0083
πππππ₯ [πππ] 6600
Table 5.1: Base case parameters for Scenario 2 simulations in the Berea network. Only
the last three parameters are varied during the sensitivity study.
89
a) Primary Drainage
We first start with the case where polar compounds remain in the oil phase (no
partitioning, π = 0), hence they do not react with the surface i.e. ππππ₯ = 0 (Equation
(5.16)). This corresponds to the conventional PD simulation at a predefined πππππ₯. The
pore occupancies are shown in Figure 5.7. Note that in the absence of wetting change,
the water saturation reaches ππ€π = 0.2, after which no more invasions are possible at
fixed πππππ₯.
Figure 5.7: Pore occupancies for the Berea network shown on the x-axis (parallel to
flow, from inlet (left) to outlet (right)) following a conventional PD at πππππ₯ = 6600 ππ
(i.e. PD/WE for the base case parameters β with π = 0 or ππππ₯ = 0Β°). The simulation
reached ππ€π = 0.2 at πππππ₯ after time π‘π = 53 πππ.
We now run the PD/WE model for the base case parameters (Table 5.1). Two effects are
opposing each other in terms of changing ππ€π following the PD/WE model. On the one
hand, the corner water collapse in oil-filled pores due to the contact angle change, as
described in Section 5.2.3(a), results in a loss in water phase connectivity. This effect,
that we call βtrappingβ, tends to increase ππ€π. On the other hand, the decrease of entry
pressures, ππππ‘ππ¦, due to the contact angle change as described in Section 5.2.3(b),
results in a gain in entry pressure accessibility. This effect, that we call βenterabilityβ,
tends to counterbalance the βtrappingβ effect by decreasing ππ€π. It follows from the
definitions that both the βenterabilityβ and βtrappingβ effects increase with ππππ₯.
The pore occupancies and wettability change following PD/WE at intermediate-wet
conditions (ΞΈmax = 80Β°) are shown in Figure 5.8. The PD/WE at ππππ₯ = 80Β° resulted in
a slightly higher ππ€π = 0.22 compared to the PD ππ€π = 0.2, meaning that the βtrappingβ
effect is slightly dominating in this case. In fact, in the intermediate-wet conditions that
we are considering (800), the drop in water connectivity is so large that the water phase
gets disconnected at ππ€ = 0.4 (ππ€ποΏ½ΜοΏ½ = 0 in Figure 5.9). Note that for ππππ₯ = 800, the
90
majority of the remaining corner water at π‘π (shown in Figure 5.8(b)-upper) must be
trapped. Indeed, most of the shapes half-angles πΎ are larger than 10Β°, thus according to
Equation (5.17), the corner films should have been expelled, if there was any outlet
connection. Note as well that only a fraction of the pores belonging to the last column
in the x-axis in Figure 5.8 represent the outlet bonds, since it also includes pores with a
range of distances from the outlet. This explains the presence of trapped water in the
corners of these particular pores. Besides, it is clear from the transition Figure 5.8(a)-
lower to Figure 5.8(b)-lower that the wettability change occurs preferentially near the
inlet at the beginning of the simulation, with the contact angle values, π, ranging from
0 to ππππ₯ . This qualitatively reproduces the early wetting alteration observed by
Bennett et al. (2004) using FESEM imaging on surfaces near the inlet. Eventually, the
adsorption steady state is reached at π‘π where a single contact angle ππππ₯ governs the
pore space.
91
Figure 5.8: Pore occupancies (upper) and (altered) contact angles (lower) for the Berea
network shown on the x-axis (parallel to flow, from inlet (left) to outlet (right))
following PD/WE for the base case parameters after (a) π‘1/2 = 31 πππ (at which ππ€ β
ππ€π+1
2) and (b) π‘π = 315 πππ (ππ€π = 0.22 at ππ
πππ₯).
92
Figure 5.9: Evolution of the oil and water phases connectivities (normalised Euler
numbers) during PD/WE for the base case parameters in the Berea network.
Effect of π½πππ on πΊππ
The pore occupancies and wettability change following PD/WE at water-wet conditions
(ΞΈmax = 30Β°) are shown in Figure 5.10. Note that at this relatively low ΞΈmax , corner
water mostly persists and is not trapped. This is shown in the pore occupancies (Figure
5.10-upper) and confirmed in Figure 5.11 as the water phase connectivity remains at its
maximum level (ΟwatΜ = 1). Consequently, the βtrappingβ effect is inhibited in this case,
which leaves the βenterabilityβ effect. Thus, quite a modest change in wetting in the
PD/WE model in this case has led to the value of Swi = 0.14 i.e. the invasion of an
additional βSo = 0.06 above the PD oil saturation. Note that the pore occupancies at
tf = 16762s (Swi = 0.14 at Pcmax) are similar to the ΞΈmax = 0Β° case shown in Figure
5.7, and the corresponding full wetting change up to ΞΈmax = 30Β° is analogous to that
showed in Figure 5.8(b)-lower.
93
Figure 5.10: Pore occupancies (upper) and (altered) contact angles (lower) for the
Berea network shown on the x-axis (parallel to flow, from inlet (left) to outlet (right))
following PD/WE for the base case parameters β with ππππ₯ = 30Β° β after π‘1/2 =
29 πππ (at which ππ€ βππ€π+1
2).
Figure 5.11: Evolution of the oil and water phases connectivities (normalised Euler
numbers) during PD/WE for the base case parameters β with ππππ₯ = 30Β° β in the
Berea network.
The dependency of ππ€π on the number of Pore Volumes (ππ) of oil injected for each
ππππ₯ value is described in Figure 5.12. It shows that each PD/WE simulation curve
(ππππ₯ > 0) is made up of two distinct portions:
An oblique line at low PV (< 1), where most of the water is displaced by oil.
94
A nearly-horizontal portion at higher PV, reaching up to 5 PV, where the decrease
in water saturation is slow. This occurs after oil break-through when the
predefined πππππ₯ is reached and the oil keeps flowing through the outlet, while
the polar compounds are resupplied from the inlet and further interact with the
surface. Consequently, the entry pressures of the remaining water-filled pores
keep deceasing and whenever one of them decreases below πππππ₯, it is invaded
by oil. This keeps going for as much PV as needed for the adsorption steady-state
to be reached (corresponding to the curves endpoints).
Figure 5.12: ππ€π as a function of ππ for the base case parameters β with different ππππ₯
imposed, in the Berea network.
The pore occupancies shown on the pore-size distribution (PSD) in Figure 5.13
demonstrate quite clearly that the PD/WE model enables the oil to reach the smallest
pores that would not have been accessible following the conventional PD (ππππ₯ = 0Β°) at
the same fixed πππππ₯.
95
Figure 5.13: Pore occupancies for the Berea network shown on the pore-size
distribution following PD/WE for the base case parameters β with (a) ππππ₯ = 0Β° (ππ€π =
0.2); (b) ππππ₯ = 30Β° (ππ€π = 0.14); (c) ππππ₯ = 60Β° (ππ€π = 0.05) and (d) ππππ₯ = 80Β°
(ππ€π = 0.22).
Effect of ππππ on πΊππ
We now keep ππππ₯ as the base case value (80Β°) and increase π€πππ₯ from 0.3 to 1.5 ππ
π2
i.e. induce a slower wetting change relative to oil invasion by requiring a higher polar
component adsorption level to alter the contact angle. It is clear from the pore
occupancy and wettability change in Figure 5.14(a) that, during the PD/WE process, the
oil front is well ahead of the wettability alteration front in the direction of flow. This
behaviour is different from that shown for the base case in Figure 5.8(a) where the two
processes are clearly more synchronised. This higher π€πππ₯ results in a slower decrease
in the water phase connectivity, as shown in Figure 5.15, with ππ€ποΏ½ΜοΏ½ = 0 at a water
saturation ππ€ = 0.14, which is significantly lower than the base case value (0.4). As a
consequence, the βtrappingβ effect is delayed and its effect is weaker. Since the
βenterabilityβ effect is insensitive to π€πππ₯, only depending on ππππ₯ , this results in ππ€π =
0.09 being significantly lower than the base case value (0.22). Note that the final
wettability distribution at tf is identical to that shown in Figure 5.8(b)-lower.
96
Figure 5.14: (a) Pore occupancies (upper) and (altered) contact angles (lower) for the
Berea network shown on the x-axis (parallel to flow, from inlet (left) to outlet (right))
following PD/WE for the base case parameters β with π€πππ₯ = 1.5 ππ
π2 β after π‘1/2 =
32 πππ (at which ππ€ βππ€π+1
2) and (b) Pore occupancies (only) at π‘π = 433 πππ (ππ€π =
0.09 at πππππ₯).
Figure 5.15: Evolution of the oil and water phases connectivities (normalised Euler
numbers) during PD/WE for the base case parameters β with π€πππ₯ = 1.5 ππ
π2 β in the
Berea network.
97
Similar to Figure 5.12, we show ππ€π as a function of network PV throughput for each
π€πππ₯ value in Figure 5.16. The behaviour is generally the same, with the presence of the
two distinctive portions at low and high ππ. Nonetheless, an exception occurred at the
lowest chosen π€πππ₯ (0.03) for which the simulation stopped early. In this case, the high
amount of trapping generated due to the fast wetting change relative to oil invasion
resulted in no further displacements being available at a relatively early stage, which is
purely due to a lack of physical accessibility, regardless of the increased entry pressure
accessibility i.e. βenterabilityβ.
Figure 5.16: ππ€π as a function of ππ for the base case parameters β with different
π€πππ₯[ ππ
π2] imposed β and the βNo Adsorptionβ case, in the Berea network.
We now aim to link the simulations to the behaviour observed by Bennett et al. (2004)
in their core-flood experiment. To do so, we focus on the evolution of the average
mobile concentration of polar compounds in the oil phase at the outlet bonds,
depending on the system adsorptive capacity, as described by Figure 5.17. For the βNo
partitioningβ case, describing a conventional PD process where polar compounds do not
partition from the oil into the water phase, these polar species emerge at the outlet at
their maximum concentration and exactly when oil breaks through (π‘ = 707π ). Besides,
when the adsorptive capacity, π€πππ₯, increases, we observe a clear delay in the
appearance of polar compounds at the outlet due to a higher surface activity.
Additionally, the high concentration at which they appear for the π€πππ₯ = 0.03 case is
mainly a contribution of the oil invasion process which carries polar compounds all
through the network in the absence of any substantial surface activity, as described in
5.2.1. For higher π€πππ₯, when the adsorptive capacity of the system is significant, the oil
98
phase appears at the outlet depleted of polar species, which are then resupplied from
the inlet through a diffusion process, as described in 5.2.2. The simulations qualitatively
reproduce the experimental trends provided in Figure 2.3(a), with the polar species
βfluoren-9-oneβ, βcarbazoleβ, βbenzocarbazoleβ and βp-cresolβ corresponding to an
increasing surface activity (i.e. higher π€πππ₯), in order. Note that each curveβs endpoint
corresponds to the adsorption steady-state being reached. At the latter, all the pores in
the network have reached maximum adsorption levels, hence maximum wetting
change, regardless of the final mobile concentrations. Ultimately, the average mobile
concentration of polar compounds in the oil phase at the outlet bonds (normalised by
πΆ0), πΆππ’π‘πππ‘Μ , would reach its maximum value, equal to 1, but this would not affect the
final contact angles and initial water saturation attained.
Figure 5.17: The evolution in time of the average mobile concentration of polar
compounds in the oil phase at the outlet bonds, normalised by πΆ0, for the base case
parameters β with different π€πππ₯[ ππ
π2] imposed β and the βNo partitioningβ case, in the
Berea network.
The pore occupancies shown on the pore-size distribution in Figure 5.18, combined with
Figure 5.13(d) (for π€πππ₯ = 0.3mg
m2), reveal that a slower wetting change relative to oil
invasion i.e. higher π€πππ₯, at intermediate-wet conditions, leads to a higher volume of
small pores being invaded following PD/WE. Indeed, the invasion process is driven
further due to a weaker βtrappingβ effect.
99
Figure 5.18: Pore occupancies for the Berea network shown on the pore-size
distribution following PD/WE for the base case parameters β with (a) π€πππ₯ = 0.03
(ππ€π = 0.42) and (b) π€πππ₯ = 1.5 ππ
π2 (ππ€π = 0.09).
Combined effect of π½πππ and ππππ on πΊππ
The combined effects of ππππ₯ and π€πππ₯ on ππ€π at fixed πππππ₯ are summarised in Figure
5.19. Note that the case ππππ₯ = 0Β°, corresponding to the conventional PD, is shown for
comparison purposes, as the notion of π€πππ₯ is not applicable in the absence of
adsorption. The results are clearly non-monotonic with regard to ππππ₯, which can be
interpreted by the competition between the two opposing effects: βenterabilityβ and
βtrappingβ, as follows:
For intermediate-wet conditions (ππππ₯ = 80Β°), where both the βtrappingβ and
βenterabilityβ effects are significant, ππ€π monotonically decreases with faster oil
invasion relative to wetting change i.e. higher π€πππ₯. Indeed, ππ€π decreases from
a value of 0.41, significantly higher than the ππππ₯ = 0Β° case (0.2), to a value as
low as 0.09. This, as explained above, is due to the βtrappingβ effect getting
delayed at higher π€πππ₯.
Switching to weakly water-wet conditions (ππππ₯ = 60Β°), leads to much lower
ππ€π, with the gap narrowing at high π€πππ₯. In fact, while both the βenterabilityβ
and βtrappingβ effects get weaker, the loss in βenterabilityβ is lower than the
decrease in βtrappingβ, meaning that the former dominates.
For water-wet conditions (ππππ₯ = 30Β°) where βtrappingβ is inhibited, ππ€π is not
sensitive to π€πππ₯. This is reasonable since by slowing down the wettability
alteration compared to the oil invasion, only the pore-filling sequence is likely to
change due to the wettability alteration. And in the absence of any generated
trapping, this does not affect the final ππ€π.
100
Finally, by uniformly distributing ππππ₯ β [0,80Β°], the resulting βππ€π vs. π€πππ₯β
curve lies between the ππππ₯ = 60 and 80Β° cases.
Note that ππ€π for both ππππ₯ = 60 and 80Β° reside between their respective slow and fast
wetting boundaries (SWB and FWB, respectively), as expected. Moreover, while the
FWB increases with higher ππππ₯ due to a more important βtrappingβ effect, the SWB
coincides for the two contact angles considered.
Figure 5.19: The resulting ππ€π as a function of π€πππ₯ [ ππ
π2] following the PD/WE model
in the Berea network for the base case parameters β with different ππππ₯ values. FWB
and SWB are the limiting fast wetting and slow wetting boundaries, respectively.
The predicted effect of the PD/WE model on conventional calculations of the oil column
saturation are now described. The ππ€ vs. height, β, above the oil water contact (OWC)
is traditionally calculated using the PD curve, and this is shown in Figure 5.20 (denoted
ππππ₯ = 0Β°). A series of simulations of the PD/WE model were carried out by varying the
main pair of parameters (ππππ₯ , π€πππ₯) at different πππππ₯ values, each corresponding to
a different height, β, in the oil column (Figure 5.20). This is a generalisation of the
previous simulations shown in Figure 5.19 that correspond to a fixed πππππ₯ i.e. a
horizontal line in the oil column (β = 3.3π). By applying the PD/WE model (ππππ₯ > 0Β°),
significant changes in phase saturations occur within the oil column, depending on ππππ₯
and π€πππ₯, as follows:
101
For ππππ₯ = 80Β°, the curves for the different chosen π€πππ₯ lie between the
boundaries FWB and SWB, with higher π€πππ₯ leading to monotonically lower ππ€π.
Additionally, the model may result in significant non-zero water saturations,
even high in the oil column. In fact, for a relatively fast wetting change (π€πππ₯ =
0.03), we observe a vertical curve. This is due to the significant amount of water
trapping being created at a relatively early stage that inhibits any further
displacement, independent of how high the attained πππππ₯ (or β) becomes.
The ππππ₯ = 60Β° curves follow the same behaviour as the ππππ₯ = 80Β° case, but
with a narrower gap between these curves as FWB is shifted to the left. Indeed,
a less important βtrappingβ effect compared to the previous case leads to ππ€π
being lower all along the oil column. Note that the two slow wetting boundaries
(SWB) for ππππ₯ = 60 and 80Β° coincide.
The ππππ₯ = 30Β° curve, which is insensitive to π€πππ₯, is shifted to the left as
compared to the conventional PD curve. In other words, PD/WE at water-wet
conditions leads to lower water saturations all along the oil column. In fact, even
at the highest point of the curve where all the pores are prone to invasion at
ππππ₯ = 0Β°, ππ€π is slightly lower due to the shrinking of water in the corners.
Figure 5.20: Distribution in the oil column in the Berea network following the PD/WE
model for the base case parameters β with different combinations of ππππ₯ and
π€πππ₯[ ππ
π2] values. FWB and SWB are the limiting fast wetting and slow wetting
boundaries, respectively.
102
Sensitivities to other parameters
Impact of πΈ
In this subsection, we assess the impact of the oil flow rate, π, on ππ€π while keeping
π€πππ₯ constant. In other words, the balance between the oil invasion and wetting change
processes is changed by varying the oil flow speed while keeping the system adsorptive
capacity constant. Note that this model is not truly dynamic but it assigns a relative time
ratio between the oil charge rate and changes in the wetting state (governed by the
adsorptive capacity).
The dependency of ππ€π on π for different ππππ₯ values is illustrated in Figure 5.21. The
qualitative behaviour is identical to the base case (see Figure 5.19). Hence, as expected,
the effect of the balance between oil invasion and wetting change on ππ€π is similar by
changing the speed of either of the processes while keeping the other one constant.
Figure 5.21: The resulting ππ€π as a function of π [ π3
π ] following the PD/WE model in
the Berea network for the base case parameters β with different ππππ₯ values. FWB and
SWB are the limiting fast wetting and slow wetting boundaries, respectively.
Impact of the invasion model
As a further sensitivity analysis, we evaluate the impact of the time-dependent invasion
model on the overall behaviour. We propose a model different from that described in
Section 5.2, for which the cumulative volume increases linearly over time. Instead, we
103
assume in this subsection that the capillary pressure, ππ , increases linearly over time:
ππ(π‘) = π β π‘ , where π‘[s] is the migration time and π is an input parameter with the
unit of ππ. π β1. We assume that a pore π is instantaneously invaded when its entry
pressure, ππππ‘ππ¦π, is overcome. However, this linear increase ceases when ππ reaches the
predefined πππππ₯, as described in Figure 5.22. Note that, at ππ = ππ
πππ₯, all the remaining
invasions become instantaneous, given that the pore is accessible, non-trapped and its
entry pressure is overcome. As in the previous model, the transport model for polar
compounds is carried out at discrete time-steps βπ‘ππ between two successive invasions.
Note that due to the adsorption of polar compounds and the resulting decrease in ππππ‘ππ¦
(see Section 5.2.3(b)), the invasion process is speeded up, especially at large contact
angles.
Figure 5.22: Illustration of the alternative time-dependent oil invasion model, similar to
that described in Figure 5.1 , but here the capillary pressure increases linearly over time
at constant Ο [ππ. π β1] until it reaches the predefined maximum capillary pressure,
πππππ₯ . Note that ππππ‘ππ¦π
denotes the entry pressure of pore π.
The resulting Swi curves as a function of π€πππ₯ for the different imposed ππππ₯ is shown
in Figure 5.23, using the base case parameters shown in Table 4.1, with the exception
that π is here replaced by π (introduced above), chosen equal to 10 ππ. π β1. It is clear
that the time-dependent capillary pressure model exhibits the same qualitative trends
as the initial model. Indeed, on the one hand, ππ€π is insensitive to π€πππ₯ for ππππ₯ = 30Β°.
On the other hand, ππ€π monotonically decreases with higher π€πππ₯ for ππππ₯ = 60Β° and
80Β°, and lies between the fast and slow wetting boundaries FWB and SWB, respectively.
Since the models are qualitatively similar, all earlier conclusions must hold.
104
Figure 5.23: The resulting ππ€π as a function of π€πππ₯ [ ππ
π2] following the PD/WE model
using an alternative time-dependent capillary pressure model in the Berea network.
The base case parameters are used β except for π replaced by Ο = 10 ππ. π β1β with
different ππππ₯ values. FWB and SWB are the limiting fast wetting and slow wetting
boundaries, respectively.
b) Waterflood
In the previous Section 5.3.1(a), we have applied the Primary Drainage/Wettability
Evolution (PD/WE) model. This has resulted in a range of final pore-scale fluid
configurations and wetting states, characterised by the main parameters of the PD/WE
model: ππππ₯; π€πππ₯ at fixed π (or vice-versa) and πππππ₯ (or β). In this section, we consider
subsequent imbibition processes by simulating water displacements in the Berea
network.
No ageing
In this first part, we simulate a waterflood in the Berea network, assuming it does not
undergo any changes in wettability other than the mild wettability alteration at ππππ₯
that occurred during primary drainage.
The pore occupancies at the end of the waterflood at fixed β = 3.3π for different ππππ₯
values are shown in Figure 5.24, corresponding to the PD/WE simulations shown in
Figure 5.13. For pores exclusively filled with water, we distinguish whether it was initially
water-filled, or invaded through snap-off or piston-like displacements. We observe a
decrease in the frequency of snap-off displacements in favour of regular piston-like
105
displacements when switching from perfectly (0Β°) to slightly less water-wet conditions
(30Β°). Further, close to intermediate-wet conditions (ππππ₯ = 60Β° and ππππ₯ = 80Β°),
snap-off is completely inhibited; this results in the mechanism of water invasion
becoming not directly dependent on ππππ₯, but rather indirectly through the initial oil
configuration (mainly πππ and the amount of corner water), which is different for each
ππππ₯.
Figure 5.24: Pore occupancies at the end of the waterflood after PD/WE for the base
case parameters β with (a) ππππ₯ = 0Β°(πππ = 0.8; πππ = 0.48), (b) ππππ₯ = 30Β° (πππ =
0.86; πππ = 0.4), (c) ππππ₯ = 60Β° (πππ = 0.95; πππ = 0.4) and (b) ππππ₯ = 80Β° (πππ =
0.78; πππ = 0.44) , shown on the pore size distribution for the Berea network (no
ageing). Note the colour key (top) for the various pore-level displacements.
More generally, each πππ obtained following PD/WE for each chosen ππππ₯ and height
β at constant π€πππ₯ (corresponding to a single data point in Figure 5.20) leads to
πππ subsequent to waterflood, as illustrated in Figure 5.25(a). The latter shows that the
change from perfectly (0Β°) to slightly less water-wet conditions (30Β°) significantly
reduces πππ, especially at high πππ , mainly due to fewer snap-off events. Indeed, snap-
106
off usually tends to considerably decrease the oil phase connectivity as the swelling of
water wetting films creates disconnected oil clusters. Although the water saturation
remaining in the corner is lower for the 30Β° contact angle due to the shrinking of the
wetting films that occurred during PD/WE, this was compensated by the strong effect of
the reduction of snap-off. Both ππππ₯ = 0Β° and 30Β° πππ curves reach a plateau at
high πππ . In contrast, for higher ππππ₯ (60Β° and 80Β°), snap-off is inhibited and water in
the corners collapsed during PD/WE. Hence, πππ monotonically decreases with
increasing πππ (at high πππ ). Actually, a surprising correlation exists for close-to-
intermediate-wet conditions between πππ and the water phase connectivity reached
following PD/WE (Figure 5.25(b)). In fact, higher πππ following PD/WE (corresponding to
higher β) results in a more disconnected water phase due to the βtrappingβ effect
discussed above. As a consequence, less βbypassingβ (introduced in 3.2.3(a)) occurs
during imbibition in favour of a more efficient microscopic sweep from the inlet. This
results in a decrease of πππ since the bypassing phenomenon has, to some extent, the
same effect as snap-off in reducing the oil phase connectivity.
107
Figure 5.25: Waterflood (a) residual oil saturations and (b) water phase connectivity
(normalised Euler number) as a function of πππ, in the Berea network, following the
application of the PD/WE model for the base case parameters β with varying height β
for the different ππππ₯ values β and no subsequent ageing (πππ€ = 0).
With ageing
We assume in this part that a predefined fraction of the pores, πππ€ , becomes oil-wet
following ageing due to the adsorption of heavier hydrocarbon species (e.g.
asphaltenes) on the rock surfaces after thin water film collapse by the mechanisms
discussed above. To achieve this, we use the physically-based Altered-Wet (AW)
distribution introduced in Chapter 3. While we uniformly distribute the oil-wet
advancing contact angles, ππ,ππ€ β [120Β°, 180Β°], we assume that the water-wet
advancing contact angles, ππ,π€π€, do not undergo any further changes after primary
drainage.
The resulting wettability distributions and pore occupancies at the end of the waterflood
at fixed β = 3.3π for different ππππ₯ values are shown in Figure 5.26 and Figure 5.27,
108
respectively, corresponding to the PD/WE simulations shown in Figure 5.13. Notice that
the slight difference in the wettability distributions for the ππππ₯ values is mainly due to
the difference in the corresponding πππ. While the invasion pattern for ππππ₯ = 30Β° is
similar to the conventional PD case (0Β°), governed by an equal extent of snap-off and
piston-like mechanisms, the pore occupancies at the end of the waterflood at
higher ππππ₯ are clearly distinct, especially for cases with the switch from 30 to 60Β°.
Indeed, if the condition in Equation (5.17) is satisfied within a pore, the corner water is
expelled if it has an outlet connection, otherwise it remains as trapped cluster. If this
pore becomes strongly oil-wet after ageing (ππ,ππ€ >π
2+ πΎ), oil films are created in the
corners following water invasion as shown in Figure 5.27(c) and (d). These are easier to
form, i.e. more abundant, than the oil layers that are sandwiched between water in the
corner and bulk (Section 3.2.4(c)). The latter are only conditionally stable according to a
strict thermodynamic criterion (refer to Section 3.2.3(d)). Additionally, same as for the
oil layers, these oil films in the corners significantly contribute in maintaining the oil
phase connectivity, hence in reducing πππ. In fact, as shown in Figure 5.28, πππ is
monotonically decreasing with higher ππππ₯ due to the increasing occurrence of corner
oil films.
Figure 5.26: Wettability distribution (AW at πππ€ = 0.5) after PD/WE for the base case
parameters β with (a) ππππ₯ = 0Β°, (b) ππππ₯ = 30Β°, (c) ππππ₯ = 60Β° and (b) ππππ₯ = 80Β°,
shown on the pore size distribution for the Berea network.
109
Figure 5.27: Pore occupancies at the end of the waterflood after PD/WE for the base
case parameters β with (a) ππππ₯ = 0Β°(πππ = 0.8; πππ = 0.54), (b) ππππ₯ = 30Β° (πππ =
0.86; πππ = 0.55), (c) ππππ₯ = 60Β° (πππ = 0.95; πππ = 0.44) and (b) ππππ₯ = 80Β° (πππ =
0.78; πππ = 0.36), shown on the pore size distribution for the Berea network (AW
distribution at πππ€ = 0.5). Note the colour key (top) for the various pore-level
displacements.
Figure 5.28: Waterflood residual oil saturations in the Berea network as a function of
πππ following the PD/WE model for the base case parameters β with varying height β
for the different ππππ₯ values β and subsequent ageing (AW distribution at πππ€ = 0.5).
110
c) Summary
PD/WE
Upon application of the developed PD/WE model on the Berea network, we reproduced
experimental patterns where an early wettability alteration occurs during primary
drainage, starting from the inlet, due to the adsorption of small polar compounds from
the oil phase. Additionally, we obtained clear differences in the PD behaviour, both at
the pore-level and in the oil column. The model particularly controlled the initial water
saturation, ππ€π, through two main parameters: (i) the level of wettability alteration (i.e.
the pre-defined maximum contact angle, πmax ) and (ii) the balance between the oil
invasion and wettability alteration processes (i.e. the system adsorptive capacity, π€πππ₯),
as follows:
Intermediate-wet conditions (ππππ₯ = 80Β°): ππ€π monotonically decreased with
faster oil invasion relative to wettability alteration i.e. higher π€πππ₯.
Weakly water-wet conditions (ππππ₯ = 60Β°): while the overall pattern was like
the 80Β° case, ππ€π was significantly lower all along the oil column.
Water-wet conditions (ππππ₯ = 30Β°): ππ€π was not sensitive to π€πππ₯ and was
lower all along the oil column compared to the conventional PD curve (ππππ₯ =
0Β°).
Subsequent waterflood
We assessed the oil recovery behaviour by carrying out a waterflood subsequent to the
PD/WE model. The residual oil saturation, πππ, value strongly depended whether a
further wettability alteration from water-wet/intermediate-wet to oil-wet conditions
was carried out during ageing. It was also dependent on the final fluid saturations and
configuration and the maximum contact angle (ππππ₯), all of which reached following
PD/WE, as explained below:
No ageing:
o Water-wet conditions (ππππ₯ = 30Β°): πππ decreased significantly
compared to the conventional PD, especially at high πππ , due to
considerably fewer snap-off displacements.
111
o Closer to intermediate-wet conditions (ππππ₯ = 60Β° and 80Β°):
πππ monotonically decreased with increasing πππ at high πππ , due to a
decreasing water phase connectivity attained following PD/WE.
With ageing:
o Water-wet conditions (ππππ₯ = 30Β°): πππ was similar to that for the
standard PD, where snap-off and piston-like displacements were equally
important.
o Closer to intermediate-wet conditions (ππππ₯ = 60Β° and 80Β°): πππ
monotonically decreased with higher ππππ₯, due to the increasing
occurrence of corner oil films.
Generally, the PD/WE model led to lower oil saturations in the subsequent waterflood.
The decrease in πππ was particularly significant when the wettability was altered from
close to intermediate-wet conditions (ππππ₯ = 60Β° and 80Β°) to oil-wet conditions during
ageing.
5.3.2 Carbonate network
In this section, we consider calculations of the type presented above, but now on the
more complex multi-scale carbonate network presented in Section 3.3.2. The base case
parameters are summarised in Table 5.2. Again, the chosen π½, πΎ, π and πΆ0 correspond
to ππππ₯ = 80Β° (Equation (5.16)). Note that we chose the πππππ₯ base case value for the
carbonate network to be significantly higher than that for the Berea network. Indeed,
the carbonate network has far smaller pores, thus requires higher capillary pressures to
achieve water saturations comparable to those presented for the Berea network. The
π€πππ₯ base case value was slightly adjusted accordingly.
112
π [π3
π β ] 5e-13
πΆ0 [ππ
πΏβ ] 500
βπ‘ππ [π ] 0.007
π 0.01
π· [ππ2
π β ] 1e-5
πΎ [πΏππβ ] 1
π€πππ₯[ππ
π2β ] 0.47
π½ 0.0083
πππππ₯ [πππ] 11713
Table 5.2: Base case parameters for Scenario 2 simulations in the carbonate network.
Only the last three parameters are varied during the sensitivity study.
a) Primary Drainage
Effect of π½πππ on πΊππ
The pore occupancies shown on the pore-size distribution after the application of
PD/WE at different ππππ₯ are shown in Figure 5.29. The PD/WE model at higher
ππππ₯ enables the oil to reach higher saturations and to invade smaller pores. In
particular, it allows oil migration into micropores (first bin of the PSD) for high enough
ππππ₯ (60 and 80Β°). Indeed, these micropores were not accessible otherwise (ππππ₯ <
60Β°) at the same fixed πππππ₯. As shown in Figure 5.30(a), the filling of micropores at
ππππ₯ = 80Β° occurred early in the process at ππ€ lower than 0.5, which coincides with
ππ€π for the conventional PD (Figure 5.29(a)). Indeed, at the same water saturation, the
filling pattern when applying PD/WE at intermediate-wet conditions compared to the
conventional PD is different as the oil invasion is more spread over the PSD. This is due
to the dramatic decrease in entry pressures following the increase in contact angles from
0Β° to ππππ₯ = 80Β° (Figure 5.30(b)), which changes the filling sequence to become less
dependent on pore size and more linked to the adsorption level of polar compounds and
the resulting wettability alteration.
113
Figure 5.29: Pore occupancies for the carbonate network shown on the pore-size
distribution following PD/WE for the base case parameters β with (a) ππππ₯ = 0Β° (ππ€π =
0.5); (b) ππππ₯ = 30Β° (ππ€π = 0.46); (c) ππππ₯ = 60Β° (ππ€π = 0.32) and (d) ππππ₯ = 80Β°
(ππ€π = 0.22).
Figure 5.30: (a) Pore occupancies and (b) wettability alteration for the carbonate
network shown on the pore-size distribution following PD/WE for the base case
parameters stopped at a predefined ππ€ = 0.5.
Effect of ππππ on πΊππ
In this part, we keep ππππ₯ as the base case value (80Β°) and alter the balance between
the processes of wetting change and oil invasion by changing π€πππ₯. The pore
occupancies shown on the pore-size distribution in Figure 5.31, added to the
information from Figure 5.29(d) (π€πππ₯ = 0.47), confirm the earlier findings that a slower
wetting change relative to oil invasion at intermediate-wet conditions results in
monotonically decreasing ππ€π at the same predefined πππππ₯. Besides, according to Figure
5.32(b), although the wettability alteration was delayed for a higher π€πππ₯ (compared to
Figure 5.30(b)), the invasion of micropores still occurred at ππ€ < 0.5. Indeed, those
invaded micropores experienced enough wettability alteration for their ππΈ to sufficiently
decrease at a relatively early stage. Eventually, oil migrates further into micropores for
higher π€πππ₯.
114
Figure 5.31: Pore occupancies for the carbonate network shown on the pore-size
distribution following PD/WE for the base case parameters β with (a) π€πππ₯ = 0.1
(ππ€π = 0.26) and (b) π€πππ₯ = 1.4 ππ
π2 (ππ€π = 0.13).
Figure 5.32: (a) Pore occupancies and (b) wettability alteration for the carbonate
network shown on the pore-size distribution following PD/WE at π€πππ₯ = 1.4 ππ
π2 and
ππππ₯ = 80Β°, stopped at a predefined ππ€ = 0.5.
Combined effect of π½πππ and ππππ on πΊππ
We summarise the combined effects of ππππ₯ and π€πππ₯ on ππ€π at fixed πππππ₯ in Figure
5.33. The results are qualitatively similar to those for the Berea network (Figure 5.19).
However, the dependency of Sπ€π on the balance between the oil invasion and
wettability alteration processes at fixed ππππ₯ is weaker for the carbonate network
because the βtrappingβ effect is less significant. In fact, as demonstrated in Figure 5.34,
the decrease in the water phase connectivity, ππ€ποΏ½ΜοΏ½, is slower for the carbonate network
at the same conditions. This pattern is attributed to the particular topology of the
multiscale carbonate network where the microporosity joins up the otherwise
disconnected larger pores. Indeed, the largest pores are generally invaded first during
PD, and many lose their corner water generating a loss in the water phase connectivity.
However, because their contribution to the overall network connectivity is low, water
remains largely connected early on, leading eventually to lower ππ€π.
115
Figure 5.33: The resulting ππ€π as a function of π€πππ₯ [ ππ
π2] following the PD/WE model
in the carbonate network for the base case parameters β with different ππππ₯ values.
FWB and SWB are the limiting fast wetting and slow wetting boundaries, respectively.
Figure 5.34: Comparison between the Berea and carbonate networksβ evolution of the
water phase connectivity (normalised Euler number) during PD/WE at ππππ₯ = 80Β° at
the highest point in the oil column for the fast wetting boundary (FWB) limiting case.
Similar to the analysis provided in Figure 5.33, we now quantitatively assess the invasion
of micropores in the carbonate network following the PD/WE model, as shown in Figure
5.35. The results clearly show that the invasion of micropores at the (base case)
moderate πππππ₯ occurs at intermediate-wet conditions (ππππ₯ = 60Β° and ππππ₯ = 80Β°).
Additionally, for ππππ₯ = 80Β°, the oil is driven further into micropores for higher Ξmax,
as expected.
116
Figure 5.35: The resulting (volumetric) fraction of micropores invaded by oil as a
function of π€πππ₯ [ ππ
π2] following the PD/WE model in the carbonate network for the
base case parameters β with different ππππ₯ values. FWB and SWB are the limiting fast
wetting and slow wetting boundaries, respectively.
More generally, by varying πππππ₯ , we obtain the oil column distributions for the
different pairs of parameters (ππππ₯, π€πππ₯) in Figure 5.36. Note that β = 6 m
corresponds to the base case πππππ₯ (11712 ππ) utilized for all previous simulations.
Again, the results of the PD/WE model follow the same qualitative trends exhibited by
the Berea network in Figure 5.20, but the gap between the curves at fixed
ππππ₯ (60 or 80Β°) is narrower since the fast wetting boundary, FWB, is shifted to the left.
This is again due to the weaker βtrappingβ effect.
117
Figure 5.36: Distribution in the oil column in the carbonate network following the
PD/WE model for the base case parameters β with different combinations of ππππ₯ and
π€πππ₯[ ππ
π2] values. FWB and SWB are the limiting fast wetting and slow wetting
boundaries, respectively.
b) Waterflood
No ageing
We simulate waterflood in the carbonate network assuming the only change in
wettability happened during PD/WE (ππππ₯). The pore occupancies at the end of the
waterflood are shown in Figure 5.37. The water invasion pattern is similar for any chosen
ππππ₯. This may be explained by the very limited snap-off for the carbonate network,
even at ππππ₯ = 0Β°. It is mainly due to a well-connected fine-scale sub-network where
the first water invasions happen, favouring piston-like displacements over snap-off. This
results in a dependency of πππ on πππ that is highly insensitive to ππππ₯ (Figure 5.38(a)),
in contrast to the Berea network case. Nonetheless, we still observe the same
correlation between πππ and οΏ½ΜοΏ½πππ‘ (reached following PD/WE) for close-to-intermediate-
wet conditions at high πππ (Figure 5.38 (a) and (b)).
118
Figure 5.37: Pore occupancies at the end of the waterflood after PD/WE for the base
case parameters β at β = 27π and (a) ππππ₯ = 0Β°(πππ = 0.98; πππ = 0.76), (b) ππππ₯ =
30Β° (πππ = 0.98; πππ = 0.74), (c) ππππ₯ = 60Β° (πππ = 0.95; πππ = 0.7) and (b) ππππ₯ =
80Β° (πππ = 0.79; πππ = 0.63) , shown on the pore size distribution for the carbonate
network (no ageing). Note the colour key (top) for the various pore-level displacements.
119
Figure 5.38: Waterflood (a) residual oil saturations and (b) water phase connectivity
(normalised Euler number) as a function of πππ, in the carbonate network, following the
application of the PD/WE model for the base case parameters β with varying height β
for the different ππππ₯ values β and no subsequent ageing (πππ€ = 0).
With ageing
In this part, we assume that ageing takes place following the PD/WE model. This process
again results in half of the pores becoming oil-wet (πππ€ = 0.5), distributed according to
the AW distribution. The wettability distributions in Figure 5.39 show that the oil-filled
micropores (defined as the first bin in the pore size distribution) are mostly oil-wet for
all cases. This is due to their tiny size that provides enough curvature for the thin films
to break (Section 4.3.2(b)). As for the Berea network, the corresponding pore
occupancies in Figure 5.40 show a significant amount of corner oil films, as well as some
oil layers, created during the waterflood subsequent to PD/WE at high ππππ₯.
The dependency of πππ on πππ in Figure 5.41 shows a pattern with a maximum at around
πππ = 0.7, regardless of ππππ₯. This is ascribed to the particular connectivity structure of
the carbonate network being mainly provided by its microporous sub-network, with the
120
larger macroporous sub-network being disconnected. Indeed, higher πππ means more
micropores filled by oil following PD/WE. And because they mostly become oil-wet after
ageing according to the AW distribution, they remain filled by water at the end of the
waterflood process. This reduces πππ as the oil maintains a much better connectivity
than the case where water-wet micropores are invaded early on, blocking the escape of
oil from the larger pores. Besides, as for the Berea network, πππ monotonically decreases
with higher ππππ₯ due to the increasing amount of the created corner oil films that
maintain the oil phase connectivity. Nonetheless, this decrease in πππ is stronger in the
carbonate network, especially at high πππ.
Figure 5.39: Wettability distribution (AW at πππ€ = 0.5) after PD/WE for the base case
parameters β at β = 27π and (a) ππππ₯ = 0Β°, (b) ππππ₯ = 30Β°, (c) ππππ₯ = 60Β° and (b)
ππππ₯ = 80Β°, shown on the pore size distribution for the carbonate network.
121
Figure 5.40: Pore occupancies at the end of the waterflood after PD/WE for the base
case parameters β at β = 27π and (a) ππππ₯ = 0Β°(πππ = 0.98; πππ = 0.3), (b) ππππ₯ =
30Β° (πππ = 0.98; πππ = 0.26), (c) ππππ₯ = 60Β° (πππ = 0.95; πππ = 0.17) and (b) ππππ₯ =
80Β° (πππ = 0.79; πππ = 0.17) , shown on the pore size distribution for the carbonate
network (AW distribution at πππ€ = 0.5). Note the colour key (top) for the various pore-
level displacements.
Figure 5.41: Waterflood residual oil saturations in the carbonate network as a function
of πππ following the PD/WE model for the base case parameters β with varying height β
for the different ππππ₯ values β and subsequent ageing (AW distribution at πππ€ = 0.5).
122
However, note that by maintaining πππ€ = 0.5 constant for each pair of parameters
(ππππ₯, β) which result in a single πππ value in Figure 5.41, Πππππ‘ is changing (refer to
5.2.3(a)). Similar to the discussion in Section 4.3.2(c), this may not be true since Πππππ‘ is
supposed to be an intrinsic property of the rock mineral and the water film. This suggests
that Πππππ‘ should be fixed as a constant, for instance equal to 33πππ (that corresponds
to πππ€ = 0.5 at ππππ₯ = 40Β°). Figure 5.42 explains the dependency of πππ€ on both ππππ₯
and β for the chosen Πππππ‘. It clearly shows, as expected, that πππ€ increases with higher
ππππ₯ since water films get more destabilised i.e. ππ β decreases in accordance with
Equation (5.18). Additionally, πππ€ increases higher in the oil column (higher β) since at
higher πππππ₯ (i) more films are prone to collapse and (ii) higher oil saturations are
reached.
Figure 5.42: Contour chart describing the relationship between ππππ₯, β and πππ€ at a
chosen Πππππ‘ = 33πππ (for which πππ€ = 0.5 at ππππ₯ = 40Β°).
c) Summary
PD/WE
Upon application of the PD/WE model, the carbonate network exhibited overall trends
qualitatively similar to the Berea network, both at the pore-scale and the reservoir-scale
(oil column). Nonetheless, the βtrappingβ i.e. loss in water phase connectivity due to the
collapse of corner water, was less effective due to the carbonate network connectivity
governed by its microporosity. This resulted in the dependency of the initial water
saturation, Sπ€π, on the balance between the oil invasion and wettability alteration
123
processes being weaker for the carbonate network, with lower Sπ€π all along the oil
column. Additionally, we clearly demonstrated how initially strongly water-wet
micropores, inaccessible with regards to their high initial capillary entry level, can be
invaded early in the primary drainage process as a result of a wetting change to
intermediate-wet conditions.
Subsequent waterflood
As for the Berea network, the residual oil saturation, πππ, upon waterflooding the
carbonate network strongly depended on whether a subsequent wettability alteration
from water-wet/intermediate-wet to oil-wet conditions was applied (ageing) following
PD/WE. Nonetheless, a few differences arose, mainly attributed to the difference in
topology:
No ageing:
πππ was weakly sensitive to the contact angle change. The main cause is the
absence of snap-off displacements in favour of more regular piston-like
displacements, probably due to a well-connected microporosity.
With ageing:
The carbonate network πππ decreased more dramatically compared to the Berea
network, especially at high πππ when micropores are filled by oil and are mainly
oil-wet in accordance with the Altered-Wet distribution. This is ascribed to the
particular connectivity of the network controlled by its microporosity.
5.4 Conclusions
In this chapter, we have developed a novel pore-scale model where the wetting state
evolves during primary drainage, referred to as the Primary Drainage/Wettability
Evolution (PD/WE) model. The model involves small polar species from the oleic phase
with high solubility in water and important surface activity (e.g. alkylphenols, carbazoles,
etc.). The PD/WE model qualitatively reproduced experimental observations reported
by Bennett et al. (2004) where an early rapid wettability alteration occurred during
primary drainage, starting from the inlet, due to the adsorption of these polar species.
The PD/WE model is proposed as a physically well-founded plausible model using all the
pore-scale physics of fluid displacements and wetting alteration that we currently
124
understand. We are applying this to generate understanding and explanations of how
the complex range of parameters interact in the primary drainage and wetting change
process that occur when crude oil invades a porous rock. The possible fluid
configurations and wetting states that can occur and how these give rise to the post-
drainage oil column are considered and explained. Furthermore, we also intended to
show that it is straightforward for oil to invade small water-filled, initially water-wet
pores with both reasonable and minimal physics-based assumptions. This final stage of
altered wettability and fluid configurations following PD/WE models forms the initial
state for the ageing and subsequent water displacement (imbibition) to determine the
oil recovery characteristics of pore networks. A flowchart describing the interactions
between the different processes modelled as part of this chapter is shown in Figure 5.43.
PD/WE
Upon the application of the wettability alteration model, we highlighted that two effects
were competing to determine the water saturation following primary drainage, ππ€π, at
fixed maximum capillary pressure, πππππ₯:
βTrappingβ: loss in water phase connectivity due to corner water removal
following the wettability alteration in oil-filled pores, which tends to increase
ππ€π. Compared to the Berea network, this effect was generally weaker in the
carbonate network due to its singular connectivity dominated by its
microporosity.
βEnterabilityβ: decrease in pore entry pressure ππππ‘ππ¦ following the wetting
change of water-filled pores, which tends to decrease ππ€π.
Interestingly, we invoked clear differences in the PD behaviour by varying both the level
of wettability alteration (through the imposed maximum contact angle reached, πmax )
and the balance between the oil invasion and wetting change processes (through the
system adsorptive capacity, π€πππ₯). These two parameters proved to dictate ππ€π
following PD/WE as they vary the competition between βtrapping and βenterabilityβ as
follows:
For intermediate-wet conditions (ππππ₯ = 80Β°), where both the βtrappingβ and
βenterabilityβ effects are significantly strong, ππ€π monotonically decreased with
faster oil invasion relative to wetting change (higher π€πππ₯) due to the resulting
125
delay in βtrappingβ. In fact, depending on π€πππ₯, the model may end up with
either lower or significantly higher ππ€π throughout the oil column compared to
the conventional PD simulation. This change in ππ€π was less dramatic for the
carbonate network than for the Berea network because of the βtrappingβ being
weaker in the former.
For weakly water-wet conditions (ππππ₯ = 60Β°), although the behaviour was
similar to the 80Β° case, ππ€π was generally lower. In fact, while both the
βenterabilityβ and βtrappingβ effects get weaker with lower contact angle, the
loss in βenterabilityβ is compensated by a stronger decrease in βtrappingβ,
meaning that the former generally dominates.
For water-wet conditions (ππππ₯ = 30Β°) where βenterabilityβ dominates in the
absence of βtrappingβ, ππ€π was independent of π€πππ₯ and was lower all along the
oil column compared to the conventional PD.
Note that neither the change in the way the balance between the oil invasion and
wettability alteration processes is varied nor the change in the time-dependent oil
invasion model has an effect on the PD/WE model qualitative behaviour.
The PD/WE model provides a physically-plausible scenario to explain the phenomenon
of oil invasion into micropores. Indeed, we demonstrated that oil did invade the
micropores at moderate capillary pressure values following their wetting alteration. In
fact, these micropores were not accessible otherwise following the conventional PD at
the same fixed πππππ₯ . This process is further illustrated in Figure 5.44.
Subsequent waterflood
In this chapter, we also carried out waterflood simulations following the application of
the PD/WE model. The resulting πππ patterns changed significantly, depending whether
a further wettability alteration from water-wet/intermediate-wet to oil-wet conditions
during ageing applied. πππ proved as well to be sensitive to the initial fluid saturations
and configuration, the extent of wettability alteration during PD/WE (ππππ₯) and the
network topology, as follows:
No ageing:
o For water-wet conditions (ππππ₯ = 30Β°), πππ was significantly lower than
the conventional PD in the Berea network, especially at high πππ where
126
a plateau was reached. This is mainly due to the occurrence of far fewer
snap-off events. However, the carbonate network lack of snap-off in
favour of more regular piston-like displacements due to a well-
connected microporosity resulted in πππ being weakly sensitive to the
contact angle change.
o For closer to intermediate-wet conditions (ππππ₯ = 60Β° and 80Β°), in the
absence of snap-off and at which some corner-water collapsed during
PD/WE, πππ monotonically decreased with increasing πππ at high πππ .
This pattern is due to the surprising correlation between πππ and the
water phase connectivity reached following PD/WE. Indeed, higher
πππ generates a less connected water phase due to βtrappingβ. This, in
turn results in less frequent βbypassingβ in favour of a more efficient
sweep from the inlet, which better maintains the oil phase connectivity
thus reduces πππ .
With ageing:
o For water-wet conditions (ππππ₯ = 30Β°), the waterflood behaviour was
like the conventional PD case, controlled to an equal extent by both
snap-off and piston-like displacements.
o For closer to intermediate-wet conditions (ππππ₯ = 60Β° and 80Β°), πππ
monotonically decreased with higher ππππ₯ , due to the creation of oil
layers and the more abundant corner films. In fact, the latter, created
during waterflood in pores whose corner water was expelled during
PD/WE and that became strongly oil-wet after ageing, are easier to form
than the oil layers that are only conditionally stable according to a strict
thermodynamic criterion. Both equally contribute in maintaining the oil
phase connectivity, hence in significantly reducing πππ. This decrease in
πππ was significantly higher for the carbonate network compared to the
Berea network, especially at high πππ when micropores are oil-filled and
largely oil-wet according to the Altered-Wet distribution.
In general terms, the PD/WE model resulted in substantially lower πππ subsequent to
waterflood, especially when the wetting state was changed from close to intermediate-
wet conditions (ππππ₯ = 60Β° and 80Β°) to oil-wet conditions during ageing. This was
particularly significant for the carbonate network, especially at high πππ when
127
micropores are invaded by oil and mostly oil-wet in accordance with the physically-
based Altered-Wet distribution. This is attributed to the connectivity characteristics of
the carbonate network where the microporosity joins up an otherwise disconnected
network.
Figure 5.43: Flowchart describing the complex interaction between the input
parameters (in orange), the modelled processes (PD/WE, Ageing and Waterflood, in
blue) and their output results (in green).
128
Figure 5.44: Representation of the PD/WE model and its effects on oil invasion by a
snap-shot of a simplified 2D 4x4 carbonate network of pores with square cross-sections
(half-angle πΎ) and two distinct pore sizes: small micropores joining up disconnected
larger macropores. Note that periodic boundary conditions apply between the bottom
and top. The network is initially water-filled and perfectly water-wet (initial contact
angle π = 0Β°), then oil displaces water from the inlet (left) to the outlet (right). For
simplicity, we assume that only the macropores can be invaded at the initial wetting
conditions and at the chosen (low) predefined maximum capillary pressure, πππππ₯ . Note
that ππππ‘ππ¦(π) of a water-filled pore corresponds to its entry pressure at π.
129
Chapter 6 : Conclusions and future work
6.1 Summary and main conclusions
In this work, we aimed at modelling the wettability alteration in microporous carbonate
rocks at the pore-scale. To do so, we investigated the responses of different
petrophysical properties to two distinct scenarios for wettability change:
Scenario 1: the wettability change occurs following primary drainage as a result
of ageing in crude oil. This corresponds to the traditional approach that mimics
the 3-stage process experienced by an initially water-wet reservoir: primary
drainage, ageing and waterflood.
Scenario 2: an alternative framework where the wettability alteration from
initially water-wet to more intermediate-wet conditions occurs during primary
drainage. A subsequent wettability alteration to oil-wet conditions may then
take place during ageing.
6.1.1 Scenario 1- Wettability alteration following ageing
What has been done
Within Scenario 1, we have developed a physically-based wettability distribution,
referred to as Altered-Wet (AW), which takes into account the shapes of the pores as
well as their size. We implemented it in a two-phase flow pore network model and used
a simple model that assigns equivalent curvatures to the flat pore walls based on the
overall pore shapes (for wettability alteration purposes only). The AW scenario
represents an alternative to the standard wettability distributions, either exclusively
correlated to pore size: Mixed-Wet Large (MWL) and Mixed-Wet Small (MWS); or
distributed independently of pore size: Fractionally-Wet (FW).
Novelty
This model is based on Kovscek et al. (1993)βs work, which has already been
implemented in 3D pore network models by many authors (Blunt, 1997, Blunt, 1998,
Oren et al., 1998, Jackson et al., 2003). However, these studies used angular pore cross
sections, and a simple parametric model for the water film collapse, assigned randomly,
which may end up in the Fractionally-Wet (FW) distribution. Moreover, to our
knowledge, neither the derived wettability distribution nor the resulting residual oil
130
saturation following waterflood have been previously compared to those corresponding
to the standard wettability distributions.
Main conclusions
The AW model qualitatively reproduced a pattern of wettability alteration in
carbonates shown experimentally by high-resolution imaging, where the most
curved pore walls are the most likely to be oil-wet.
The AW distribution was different from any of the standard wettability
distributions (MWL, MWS, FW). Particularly in the carbonate network, although
it led to mostly oil-wet micropores due to their very small sizes (which provide
enough curvature for the water films to collapse), the wetting distribution was
still clearly distinct from the MWS model.
The sandstone network showed little sensitivity to different wettability scenarios
with regard to the residual oil saturation.
The microporous carbonate network exhibited a significant effect of the
wettability distributions on the residual oil saturation.
We demonstrated the importance of the microporesβ wettability on flow in
carbonates, which proved to control the oil recovery.
6.1.2 Scenario 2- Wettability alteration starting during primary drainage
What has been done
Within Scenario 2, based on experimental observations by Bennett et al. (2004), we
developed a physically plausible wettability alteration model that occurs during primary
drainage. The model, implemented in a two-phase flow pore network model, involves
the relatively rapid adsorption of surface-active water-soluble polar compounds (such
as phenols, carbazoles, etc.) present in the oleic phase. This then leads to a
destabilisation of the water films (by reducing the disjoining pressure) leading to wetting
alteration by the subsequent adsorption of higher molecular weight hydrocarbon
species (e.g. asphaltenes).
Novelty
This model proposes a novel scenario that tackles the issue of oil migration into
micropores by associating it to a wettability change that evolves during primary drainage
(PD). Moreover, the approach itself is original since we overcame the time-
131
independency inherited from βquasi-staticβ pore network models by adding a notional
time-dependency to the conventional oil invasion in order to incorporate a dynamic
transport model for polar compounds. However, although the model is based on
experimental evidence, we believe that it is not yet at a predictive stage, but rather an
explanative model based on reasonable assumptions.
Main conclusions
The model qualitatively reproduced experimental observations where an early
rapid wettability alteration occurred during primary drainage, starting from the
inlet, due to the polar compounds surface interaction that decreased in the
direction of flow.
We invoked clear differences in the primary drainage patterns by varying the
level of wettability alteration and the balance between the oil invasion and
wetting change processes. In fact, these two parameters proved to dictate ππ€π.
For the particular conditions of relatively fast oil invasion or small wetting
changes, the model leads to lower ππ€π at fixed πππππ₯ compared to the
conventional primary drainage model. Particularly in the microporous carbonate
network, this results in the invasion of micropores at reasonable (low) capillary
pressures.
The model resulted in significant decrease in the residual oil saturations, πππ ,
following subsequent waterflood, compared to the conventional PD. This
improvement in oil recovery was especially significant when an additional
wettability alteration from intermediate-wet to oil-wet conditions applies during
ageing.
6.2 Discussion
Although the two scenarios developed in the course of this study particularly
served our purpose in reproducing experimental wettability alteration trends
observed in microporous carbonates and in providing a possible explanation of
micropores invasion, they apply regardless of the rock type. The only parameter
that needs to be modified when changing from a rock to another would be the
critical disjoining pressure, which is for instance lower for carbonates than for
sandstones.
132
The transport model incorporated in Scenario 2 added a significant complexity
to the conventional quasi-static pore network modelling tool. Hence, the
computation times (shown in Table 6.1) have considerably increased.
Nonetheless, they are still within a reasonable time-scale of a few hours in our
case, which can be improved by a more powerful computer. This computational
complexity is mainly attributed to the discretised diffusion model that requires
small diffusion time-steps in order to conserve mass properly (see Equation
(5.9)). Since this constraint is imposed by the smallest pores, the wider is the
pore-size distribution, the slower is the simulation. This explains the higher
computational time for the carbonate network whose pore sizes span multiple
length-scales, as compared to the Berea network with comparable number of
pores. Consequently, we expect that complex multi-scale rocks will require larger
networks to produce reliable quantitative predictions.
Scenario 1 Scenario 2
Berea network π‘ = 36π π‘ = 53π
Carbonate network π‘ = 17994π π‘ = 32986π
Table 6.1: Typical computational times for the two developed scenarios run on the
Berea and carbonate networks using the base case parameters (shown in Table 4.1,
Table 5.1 and Table 5.2). Note that the computer has an Intel core i7 processor, and
that multiple simulations can be run simultaneously without altering the computational
efficiency
We attempted to run our simulations on a different multiscale carbonate
network with significant microporosity and where the macropores are (just)
intra-connected. However, we were not able to identify a suitable network. Most
of the multiscale networks that we generated lacked the overall connectivity to
allow the waterflood to reach reasonable residual oil saturations (lower than 1).
The reason possibly resides in the shortcomings of multiscale network extraction
techniques that fail to properly join the networks at two distinct length-scales.
133
6.3 Future work
We acknowledge that the models that we developed in the scope of this thesis have
shortcomings and issues that need to be addressed and require further improvements,
such as:
Within the framework of Scenario 1:
o Implement Grain Boundary Pore (GBP) cross-sectional shapes (Man and
Jing, 2000) within the pore network model instead of assigning
equivalent curvatures to flat pore shapes.
o Better characterise the micropores shapes, for instance capture the
distinctive straight and platy morphology of micropores (Cantrell and
Hagerty, 1999).
o Consider a wider dataset of carbonate networks as inputs to the model
to support the findings.
Within the framework of Scenario 2:
o We implemented a simple transport model that sufficiently captured
some interesting primary drainage patterns. However, we suggest the
implementation of more sophisticated chemical and physical details into
the diffusion, adsorption and subsequent wettability change sub-
models. For instance, does the Langmuir isotherm for equilibrium
adsorption capture the actual adsorption mechanism or should we
rather use a kinetic adsorption model instead? Additionally, one can
suggest a more complex model for wettability change as a function of
adsorption levels instead of the rather simple linear dependency that we
proposed.
o We stretched the capabilities of βquasi-staticβ network models by adding
a notional time-dependency in order to incorporate the transport model.
Hence, we suggest to assess the benefit of using a truly βdynamicβ pore
network model instead. However, this should capture accurately films in
the corners as these proved to be central in controlling the resulting
residual phase saturations (ππ€π and πππ) in our model. Moreover,
dynamic models are computationally very demanding, therefore
134
benefits of using a dynamic model should be looked at critically,
particularly if the considered flow regimes are still capillary dominated.
o In order to run the model on bigger networks, more representative of
real pore spaces, there is a need to address the computational issues
related to the discretised diffusion model. This is especially relevant for
multiscale networks that possess a wide range of pore sizes.
o We would also propose additional core-flood experiments be carried out
to confirm our findings. The essence of such experiments could be similar
to the experiments conducted by Bennett et al. (2004), but using two
different crude oils: one which is rich and the other which is poor in small
polar compounds such as alkylphenols, carbazoles, etc.. The two
resulting oil/water saturations would then be compared, and our
prediction that these should be significantly different could be tested.
We further suggest a more dynamic assessment of the wettability
change, using for example in-situ contact angle measurements directly
from micro-CT images (Andrew et al., 2014), to confirm the evolution of
the wettability change during primary drainage. Additionally, in order to
test our prediction of the removal of water films in the pore
corners/crevices due to the evolving wettability change, we propose the
evaluation of the remaining water at the pore-scale using micro-CT
imaging as described by Pak et al. (2015).
135
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